NAVIGATION 

AND 

NAUTICAL  ASTRONOMY 

INCLUDING 

THE  THEORY  OF  COMPASS  DEVIATIONS 


A  TREATISE  ON 

NAVIGATION 

AND 

NAUTICAL  ASTRONOMY 

INCLUDING 

THE  THEORY  OF  COMPASS  DEVIATIONS 

PREPARED    FOR    USE   AS   A   TEXT-BOOK 

AT   THE 

U.  S.  NAVAL  ACADEMY 

BY 

COMMANDER  W.  C.  P.  MUIR,  U.  S.  NAVY 

Head    of    the   Department    of    Navigation 
U.  S.  Naval  Academy 


THIRD  EDITION 
Revised  and  En'argeu 


ANNAPOLIS,  MARYLAND 

THE  UNITED  STATES  NAVAL  INSTITUTE 

1911 


COPYRIGHT,  1906 
COPYRIGHT,  1908 
COPYRIGHT,  1911 

BY 

PHILIP   R.    ALGER 

Secretary  and  Treasurer 
U.  S.  NAVAL  INSTITUTE 


£ot&  (gafcttnore 

BALTIMORE,  MD.,  U.  8.  A. 


M^ 


PEEFACE. 

In  this  volume,  the  endeavor  has  been  made  to  place  under 
one  cover  the  allied  subjects  of  Navigation,  Theory  of  Com- 
pass Deviations,  and  Nautical  Astronomy;  and,  though  the 
book  has  been  written  primarily  for  the  use  of  midshipmen, 
it  is  believed  that  the  various  subjects  have  been  so  presented 
that  any  zealous  student  with  only  a  slight  knowledge  of 
trigonometry  may  be  able  to  master  any  method  given. 

An  effort  has  been  made  to  include  in  this  work  not  only 
the  results  of  a  large  practical  experience  at  sea,  but  informa- 
tion gleaned,  during  tours  of  duty  in  this  department, 
from  a  study  of  the  best  English  and  American  authorities. 
Much  attention  has  been  given  to  a  description  of  the  various 
navigational  instruments,  their  uses,  and  errors;  and  to  the 
principles  involved  in  the  construction  of  charts  as  well  as  to 
an  account  of  the  work  usually  performed  on  them. 

The  Theory  of  the  Deviations  of  the  Compass  has  been 
presented  in  the  popular  way  for  the  practical  man,  and  from 
a  mathematical  standpoint  for  more  advanced  students. 

In  Part  II  enough  of  Theoretical  Astronomy  has  been  in- 
corporated to  enable  any  one  without  a  previous  knowledge 
of  that  science  to  pursue  the  study  of  the  practical  part  of 
Nautical  Astronomy. 

In  the  chapter  on  Time,  I  have  gone  much  into  detail  and 
have  illustrated  the  theory  by  the  solutions  of  many  examples, 
because  an  experience  of  years  as  an  instructor  has  shown 

272623 


vi  PREFACE 

that  beginners  usually  find  it  an  intricate  subject.  In  this 
chapter,  as  in  all  other  parts  of  the  book,  practical  illustra- 
tions follow  immediately  the  theory  on  which  they  are  based. 

In  a  consideration  of  "  lines  of  position,"  much  space  has 
been  given  not  only  to  the  theories  and  practice  of  Sumner, 
but  also  to  the  later  adaptation  of  those  theories  by  Johnson 
and  Marcq  Saint-Hilaire.  All  these  methods  are  worthy  of 
close  examination,  and  each  will  be  found  to  have  its  special 
advantages.  However,  if  the  student  is  pressed  for  time,  he 
is  advised  to  confine  his  attention  to  what  will  be  described 
as  the  "  chord  method  "  which  embodies  the  present  practice 
of  the  United  States  Naval  Service. 

A  chapter  on  "  Tides  "  and  one  on  "  The  Identification  of 
Heavenly  Bodies  "  have  been  included ;  a  knowledge  of  both 
these  subjects  is  essential  to  the  modern  navigator. 

The  text  contains  no  reference  to  "  lunars,"  that  method  of 
finding  longitude  being  regarded  as  obsolete  in  these  days  of 
excellent  chronometers.  It  is  believed  that  the  time  is  not 
far  distant  when  "  lunar  distances  "  will  disappear  from  the 
Nautical  Almanac. 

Acknowledgments  are  due  to  Lt.-Comdr.  W.  V.  Pratt, 
IT.  S.  N.,  and  to  Lt.  C.  P.  Snyder,  IT.  S.  N.?  for  assistance  in 
proof-reading;  to  Midshipman  H.  G.  Knox,  U.  S.  N.,  for 
work  on  Tables  II  and  III ;  and  to  the  following-named  firms 
for  the  loan  of  certain  electrotypes :  E.  S.  Eitchie  &  Sons  of 
Boston,  Keuffel  &  Esser,  T.  S.  &  J.  D.  Negus,  and  John  Bliss 
&  Co.  of  New  York. 

In  conclusion,  I  desire  to  express  my  gratitude  to  Lt.- 
Comdr.  B.  W.  Wells,  U.  S.  N.,  and  to  Lt.-Comdr.  G.  K. 


PREFACE  vii 

Marvell,  U.  S.  1ST.,  for  valuable  criticism  of  the  original 
manuscript  and  for  assistance  in  eliminating  errors  from  the 
finished  book. 

W.  C.  P.  MUIR. 
DEPARTMENT  OF  NAVIGATION, 
U.  S.  NAVAL  ACADEMY,  MAY  1,  1906. 

NOTE  TO  SECOND  EDITION. 

Chapter  XX  has  been  rewritten  and  enlarged,  and,  besides 
a  few  minor  changes  which  have  been  made  in  certain  parts 
of  the  text  where  deemed  desirable  for  greater  clearness,  the 
following  additions  have  been  made :  a  new  method  of  equal 
altitudes  suggested  by  Mr.  G-.  W.  Littlehales,  Hydrographic 
Engineer,  U.  S.  Navy  Department;  plates  X  to  XVI;  and 
four  appendices. 

This  opportunity  is  taken  of  thanking  those  officers  of  the 
service  and  friends  outside  the  service  who  have  kindly  ex- 
pressed appreciation  of  my  efforts,,  or  who  may  have  offered 
suggestions  with  reference  to  this  edition. 

W.  C.  P.  M. 

JULY  1,  1908. 

NOTE  TO  THIRD  EDITION. 

A  complete  revision  of  this  book  has  been  made  wherever 
changes  have  been  found  necessary  to  make  the  text  and  ex- 
amples correspond  with  the  new  method  of  estimating  course 
and  azimuth,  an  innovation  due  to  the  recently  adopted  grad- 
uation of  the  compass.  Some  new  matter  has  also  been  added. 

For*suggestions  as  to  this  revision,  my  thanks  are  due  Com- 
manders S.  S.  Eobison  and  J.  H.  Sypher,  U.  S.  N.,  and  to 
Commander  Geo.  R.  Marvell,  U.  S.  N.,  who  relieved  me  as 
Head  of  Department  of  Navigation,  U.  S.  Naval  Academy,  on 
my  detachment  from  active  duty,  August,  1909. 

W.  C.  P.  M. 
SHELBYVILLE,  KY.,  MARCH  15,  1911. 


LIST  OF  WOEKS  CONSULTED. 

Spherical  and  Practical  Astronomy,  Chauvenet. 

General  Astronomy,  Young. 

Elements  of  Astronomy,  White. 

The  Heavens,  Guillemin. 

Navigation  and  Nautical  Astronomy,  Coffin. 

Navigation,  Asa  Walker. 

American  Practical  Navigator,  Bowditch. 

Navigation  and  Nautical  Astronomy,  Martin. 

Navigation,  Merrifield  &  Evers. 

Nouvelle  Navigation  Astronomique,  Villarceau. 

Modern  Navigation,  Hall. 

Wrinkles  in  Practical  Navigation,  Lecky. 

Finding  a  Ship's  Position  at  Sea,  Sumner. 

Finding  Latitude  and  Longitude  in  Cloudy  Weather,  A.  C.  John- 
son. 

How  to  Find  the  Stars,  Rosser. 

Handy  Book  of  the  Stars,  Whall. 

Marine  Surveying,  IL,  S.  N.  A. 

Notes  on  Navigation,  U.  S.  N.  A. 

Practical  Problems  and  Compensation  of  the  Compass,  Diehl. 

British  Admiralty  Manual  of  Deviations. 

Mathematical  Theory  of  the  Deviations  of  the  Compass,  Howell. 

Manual  of  Deviatons,  F.  J.  Evans. 

Instructions  for  Care  of  Chronometers  and  Watches,  Bureau  of 
Equipment,  Navy  Department. 

Proceedings  U.  S  Naval  Institute. 

Method  of  Least  Squares,  Merriman. 


CONTENTS. 

PAET  I.     NAVIGATION  AND  COMPASS  DEVIATIONS. 
CHAPTER  I. 

PAGE 

General  definitions:  Navigation,  pilotage,  nautical  astronomy, 
observations  on  the  form  and  size  of  the  earth. — Axis. — 
Poles. — Equator. — Meridians. — The  Prime  meridian. — Par- 
allels of  latitude. — Latitude, — Difference  of  latitude. — 
Middle  latitude. — Longitude. — Difference  of  longitude. — 
Geographical  and  nautical  miles. — Rhumb  line  or  loxo- 
dromic  curve. — Course. — Distance. — Bearing  of  an  object 
or  place  1-5 

CHAPTER  II. 

List  of  navigational  instruments  and  books  usually  provided. 
— Speed  measures:  Revolution  of  screw;  patent  logs; 
log  chip,  line  and  sand  glass. — Sounding  apparatus: 
Hand,  coasting,  and  deep  sea  leads;  Thompson's  and 
Tanner's  sounding  machines. — Use  of  sounding  data. — 
Charts:  description,  construction,  and  use  of  polyconic, 
polar',  gnomonic,  and  Mercator  charts;  advantages  and 
disadvantages  of  each  system  of  projection;  conventional 
notation  and  hydrographic  signs;  plotting  and  taking  off 
positions,  laying  courses,  and  measuring  distances;  cor- 
rection of  charts;  arrangement  and  stowage  on  board. — 
The  3-arm  protractor  and  its  use. — The  "  Three-point 
problem  "  and  rules  governing  the  selection  of  objects  to 
be  angled  on 5-44 

CHAPTER  III. 

Section  I.  The  compass. — The  U.  S.  Navy  compensating  bin- 
nacle.— The  azimuth  circle. — The  pelorus. — The  illumi- 
nated dial  pelorus. — The  use  of  a  pelorus  to  determine  a 
magnetic  heading  44-55 

Section  II.  The  earth's  magnetism  and  the  elements  of  that 
magnetism. — Magnetic  poles. — Magnetic  equator. — Mag- 
netic meridian. — Magnetic  latitude. — Relation  of  true  and 
magnetic  meridians. — Variation. — Deviation. — Correction 
of  bearings. — Leeway. — Correction  of  courses. — Local  at- 
traction   55-66 


CONTENTS 


PAGE 

Section  III.  Finding  the  deviation  by  reciprocal  bearings,  by 
bearings  of  a  distant  object,  by  time  azimuths,  by  ranges. 
— Description,  construction,  and  use  of  Napier's  diagram .  66-77 

Section  IV.  "  Hard  "  and  "  soft "  iron. — Magnetic  induction. 
— How  a  ship  becomes  magnetic. — Magnetic  forces  acting 
on  a  compass  needle  in  an  iron  or  steel  ship  and  the 
effect  of  each  in  producing  deviation. — Semicircular  de- 
viation.— Quadrantal  deviation. — Constant  deviation. — 
Causes  and  characteristics  of  each  kind  of  deviation. — 
The  approximate  equation  for  deviation. — Determination 
of  the  approximate  coefficients  by  inspection  of  a  devia- 
tion table. — Heeling  error. — Mean  directive  force.. 77-104 

CHAPTER  IV. 

Section  I.  Mathematical  theory  of  compass  deviations. — 
Consideration  of  the  various  forces  acting  on  a  compass 
needle  in  an  iron  or  steel  ship. — Finding  the  components 
of  each  force  in  certain  definite  directions  through  the 
compass  and  then  the  resultant  of  all  the  forces  in  each 
direction. — Representation  of  the  effect  of  the  soft  iron 
of  the  ship  by  nine  soft  iron  rods. — Symmetrical  and  un- 
symmetrical  soft  iron. — The  fundamental  equations . . .  104-116 

Section  II.  Transformation  of  the  fundamental  equations. — 
Forces  of  earth  and  ship  to  head,  to  starboard,  to  mag- 
netic North,  and  to  magnetic  East. — Formulae  for  com- 
puting deviations. — Subdivisions  of  the  deviation  and  a 
consideration  of  the  various  coefficients. — The  ship's  polar 
force  and  starboard  angle. — The  "  Gaussin  error  ". . . .  .116-128 

Section  III.     Method  of  least  squares  applied  to  the  determi- 
nation  of   coefficients. — Formation   of  normal  equations. — 
Equations  for  exact  coefficients  in  terms  of  the  approxi- 
mate coefficients   128-136 

Section  IV.  Analysis  of  deviations  and  the  determination  of 
exact  coefficients  136-142 

Section  V.  Observations  for  horizontal  and  vertical  forces 
ashore  and  on  board. — Determination  of  X  and  fi 142-150 

Section  VI.  Determination  of  33,  (£,  and  2)  by  observations  in 
one  quadrant,  in  one  semicircle. — Determination  of  S3  and 
(£  from  observations  of  deviation  and  horizontal  force  on 
one  heading;  of  39,  GT,  3D,  X,  a,  and  e  from  observations  on 
two  headings. — Determination  of  the  forces  of  hard  and 
soft  iron  causing  semicircular  deviation. — Placing  a 
Flinders  bar. — Computation  of  deviations  from  coeffi- 
cients  150-165 

Section  VII.  Heeling  error. — Change  in  the  fundamental 
equations  due  to  the  ship's  heeling. — The  "  Heeling  co- 
efficient."— Methods  of  determining  heeling  error. — Cor- 
rection of  heeling  error  by  vibrations,  by  using  the  heel- 
ing adjuster,  by  the  tentative  method 165-178 


CONTENTS  xi 


PAGE 

Section  VIII.  Compensation  of  the  compass:  (1)  when  devia- 
tions are  known,  (2)  when  deviations  are  unknown. — De- 
termination of  magnetic  courses  when  the  deviations  are 
unknown,  using  the  pelorus  or  azimuth  circle. — Given 
93,  (£,  and  S),  to  compensate  on  one  heading 178-189 

Section  IX.  The  dygogram:  its  construction  and  use. — To 
construct  a  dygogram  when  the  exact  coefficients  are 
known. — To  construct  a  dygogram  when  the  deviations 
and  horizontal  force  are  known  (1)  for  two  opposite  mag- 
netic courses,  (2)  for  two  magnetic  courses  not  oppo- 
site   o 189-204 

CHAPTER  V. 

Piloting. — The  bearing  of  an  object. — A  line  of  position. — A 
line  of  bearing. — A  position  point. — Fixing  ship's  position 
near  land  (1)  by  sextant  angles;  (2)  by  cross  bearings; 
(3)  by  a  bearing  and  distance;  (4)  by  change  of  bearing 
of  a  single  object,  using  tables  or  the  graphic  method; 
(5)  by  doubling  the  angle  on  the  bow. — Distance  of  pass- 
ing an  object  abeam.: — Horizontal  and  vertical  danger 
angles. — Danger  bearings. — Lights  as  danger  guides. — 
Fog  signals  204-220 

CHAPTER  VI. 

The  sailings. — Preliminary  definitions. — Plane  sailing. — Us6 
of  traverse  table. — Traverse  sailing.  Graphic  explanation 
of  traverse  sailing. — Sources  of  data. — Preparation  of  the 
traverse  form  and  data. — Parallel  sailing. — Middle  lati- 
tude sailing. — When  not  advisable  to  use  middle  latitude 
sailing. — Current  sailing:  solutions  by  construction,  by 
trigonometry,  and  by  the  traverse  table. — Mercator  sail- 
ing.— Graphic  illustration  of  the  theory  of  Mercator  sail- 
ing.— When  not  advisable  to  use  Mercator  sailing. — Cor- 
rection to  the  middle  latitude. — Day's  work  by  D.  R..  .220-263 

CHAPTER  VII. 

Great  circle  sailing. — Comparison  of  rhumb  and  great  circle 
tracks. — Preliminary  definitions. — Great  circle  course  and 
distance  by  (1)  computation,  (2)  azimuth  tables,  (3) 
great  circle  charts,  (4)  graphic  approximation. — Finding 
the  vertex  and  point  of  maximum  separation. — Use  of  ter- 
restrial globe. — Graphic  chart  methods. — Composite  sail- 
ing by  (1)  gnomonic  charts,  (2)  computation,  (3)  graphic 
methods  . .  263-285 


xii  CONTENTS 

PART  II.    NAUTICAL  ASTRONOMY. 
CHAPTER  VIII. 

PAGE 

General  definitions:  Nautical  astronomy,  the  celestial  sphere, 
axis,  poles,  celestial  equator,  horizons,  zenith,  nadir,  ce- 
lestial meridian,  ecliptic,  equinoctial  and  ecliptic  points, 
the  colures. — Consideration  of  spherical  coordinates. — The 
ecliptic  system:  celestial  latitude  and  longitude. — The 
equinoctial  system:  declination,  polar  distance,  parallels 
of  declination,  hour  circles,  transit,  hour  angle,  solar  time, 
sidereal  time,  relation  between  solar  and  sidereal  days, 
right  ascension,  relation  of  hour  angle  and  right  ascen- 
sion.— The  horizon  system:  celestial  horizon,  vertical 
circles,  prime  vertical,  azimuth,  amplitude,  altitude, 
zenith  distance. — Proof  that  latitude  equals  the  altitude 
of  the  elevated  pole. — The  astronomical  triangle. — Projec- 
tions on  planes  of  the  meridian,  the  equator,  and  hori- 
zon   , . .  287-303 

CHAPTER  IX.       ' 

The  sextant. — The  optical  principle  of  the  sextant  and  its  ap- 
plication in  the  measurement  of  angles. — The  vernier. — 
Reading  the  sextant. — Excess  of  arc. — Constant  and  acci- 
dental errors  of  the  sextant. — Errors  of  graduation  and 
eccentricity. — Prismatic  effect  of  mirrors  and  shade 
glasses. — Adjustment  of  the  sextant. — Index  error  and  its 
determination. — Using  a  sextant  to  observe  altitudes  of 
the  sun,  or  a  star,  at  sea. — Measuring  horizontal  angles. — 
General  care  of  sextant. — Resilvering  mirrors. — The  arti- 
ficial horizon,  its  care  and  preparation  for  use,  its  ad- 
vantages, its  theory. — Method  of  observing  with  artifi- 
cial horizon 303-327 

CHAPTER  X. 

Description  of  chronometers. — Reception  and  stowage  on 
board. — Winding  and  comparison. — Cleaning  and  oiling. — 
Transportation. — Effect  of  change  of  temperature. — Hart- 
nup's  law. — General  equation. — Temperature  curves. — In- 
structions for  management  and  use. — Comparing,  stop,  and 
torpedo-boat  watches. — Winding,  general  care,  and  prepa- 
ration for  shipment 327-340 

CHAPTER  XL 

Comparison  of  sidereal  and  tropical  years. — The  calendar. — 
The  Gregorian  correction. — The  sidereal  year. — Relation 
of  solar  and  sidereal  time..  ..340-343 


CONTENTS  xiii 

CHAPTER  XII. 

PAGE 

Time  and  its  measurement. — Sidereal  day. — Apparent  solar 
day. — Mean  solar  day. — Equation  of  time. — Relation  of 
local  sidereal  time,  the  hour  angle,  and  right  ascension 
of  a  body. — Astronomical  and  civil  time. — Standard  time. 
— Conversion  of  arc  into  time  and  vice  versa. — Relation 
between  the  local  times  at  two  places. — Finding  Green- 
wich time. — Gain  or  loss  of  time  with  change  of  position. 
— Crossing  the  180th  meridian 343-362 

CHAPTER  XIII. 

The  Nautical  Almanac. — Finding  a  required  quantity  for  a 
given  time. — Use  of  second  differences. — Mean  time  of 
moon's  transit. — Mean  time  of  a  planet's  transit 362-380 

CHAPTER  XIV. 

Interconversion  of  apparent  and  mean  times. — Formulae  for 
the  interconversion  of  mean  and  sidereal  time  intervals. — 
Conversion  of  mean  time  into  sidereal  time  and  vice 
versa. — Conversion  of  apparent  time  into  sidereal  time 
and  vice  versa. — Relation  of  time,  hour  angles,  and  right 
ascensions,  and  problems  involving  them. — To  find  the 
local  mean  time  of  upper  transit  of  a  particular  heavenly 
body,  also  the  time  of  lower  transit. — To  find  the  watch 
time  of  transit  of  the  sun,  of  a  star,  the  moon,  or  a  planet. 
— To  find  the  hour  angle  of  any  heavenly  body  at  a  given 
time  and  place. — To  find  what  stars  will  cross  the  merid- 
ian between  two  given  times 380-411 

CHAPTER  XV. 

Corrections  to  an  observed  altitude,  using  (1)  a  sea  horizon, 
(2)  an  artificial  horizon. — Refraction. — Parallax. — Dip  of 
the  horizon. — Error  of  dip. — Distance  of  sea  horizon. — 
Range  of  visibility  at  sea. — Dip  of  a  point  nearer  than  the 
sea  horizon. — Apparent  semi-diameter. — Augmentation  of 
the  moon's  semi-diameter. — Theoretical  and  practical 
methods  of  correcting  altitudes. — Correction  of  altitude 
for  run  411-432 

CHAPTER  XVI. 

Solution  of  the  "astronomical  triangle." — Finding  the  true 
altitude. — Altitude  and  time  azimuth  of  a  heavenly  body. 
— Altitude-azimuth  of  a  heavenly  body.— The  haversine 
formula  for  azimuth. — Amplitudes. — Use  of  azimuths  and 
amplitudes  in  finding  compass  error. — To  determine  when 
an  error  in  altitude,  or  an  error  in  latitude,  will  have 
the  least  effect  on  the  azimuth. — True  bearing  of  a  ter- 
restrial object. — Hour  angle  and  local  time  from  an  ob- 
served altitude. — Haversine  formula  for  hour  angle. — 


xiv  CONTENTS 


PAGE 

Conditions  of  observation. — Sunrise  or  sunset  sights. — 
Time  of  sunset. — Duration  of  twilight. — Hour  angle  of 
any  heavenly  body  when  in  the  horizon. — Length  of  day 
and  night. — To  determine  when  an  error  in  altitude,  or 
an  error  in  latitude,  will  have  the  least  effect  on  the  hour 
angle,  and  hence  the  best  time  to  observe  for  longitude. — 
Hour  angle  of  a  heavenly  body  when  on  or  nearest  to 
the  prime  vertical 432-488 

CHAPTER  XVII. 

Latitude  from  meridian  altitudes  above  or  below  the  pole. — 
The  constant. — Finding  latitude  by  observations  of  bodies 
out  of  the  meridian:  (1)  The  0"0'  method,  (2)  an  ap- 
proximate method  involving  both  latitude  and  longitude, 
(3)  by  reduction  to  the  meridian,  (4)  by  altitude  of  Po- 
laris, (5)  Chauvenet's  method,  (6)  Prestel's  method. — 
"  Angle  of  the  vertical "  or  "  reduction  of  the  lati- 
tude "  488-548 

CHAPTER  XVIII. 

Chronometer  error. — Distinction  between  error  and  correc- 
tion.— To  find  the  rate. — Sea  rate. — Irregular  rate. — 
Finding  error  and  rate  by  transits;  by  time  signals; 
single  or  double  altitudes;  equal  altitudes  of  a  fixed  star, 
the  sun,  or  a  planet. — Rating  chronometers  by  telegraph 
or  wireless  signals. — The  U.  S.  system  of  time  signals. — 
To  correct  the  middle  time  in  equal  altitudes  for  a  small 
difference  of  altitude. — Methods  of  observation. — Com- 
parison of  equal  and  double  altitudes. — Longitude  ashore 
by  electric  signals;  ashore  or  afloat  by  equal  altitudes, 
Bingle  altitudes,  double  altitudes. — An  approximate  meth- 
od of  equal  altitudes  of  the  sun  for  longitude  at  sea. — 
A  method  of  finding  longitude  at  sea  by  equal  altitudes 
of  the  sun  when  the  ship  is  proceeding  at  high  speed. .  .548-589 

CHAPTER  XIX. 

Sumner's  method. — A  heavenly  body's  geographical  position 
and  its  coordinates. — Circles  of  equal  altitude. — A  line  of 
position. — Curves  of  equal  altitude  on  a  Mercator  chart. 
— Determination  of  points  on  the  curve. — Double  alti- 
tude problem. — Simultaneous  observations. — Advantages 
of  simultaneous  over  double  altitude  observations. — Re- 
lation between  circles  of  equal  altitude  and  the  astro- 
nomical triangle. — Method  of  determining  a  line  of  posi- 
tion.— Graphic  illustrations  of  the  various  ways  in  which 
a  line  of  position  may  be  used  at  sea  or  near  the  coast. — 
To  allow  for  uncertainties  in  time  or  altitude. — Defini- 
tion of  longitude  and  latitude  factors. — Plotting  lines  by 
the  chord  method,  by  the  tangent  method. — Position  by 


CONTENTS  xv 


PAGE 

simultaneous  observations,  one  body  on  the  meridian, 
one  on  or  near  the  prime  vertical. — Position  by  the  "  mu- 
tual correction  "  method. — Computing  the  intersection  of 
two  lines  determined  (1)  by  the  chord  method,  (2)  by 
the  tangent  method 589-636 

CHAPTER  XX. 
The  new  navigation  or  the  method  of  Marcq  Saint-Hilaire. 636-661 

CHAPTER  XXI. 

A  day's  work  at  sea,  rules,  form,  and  solution  of  an  ex- 
ample   661-670 

CHAPTER  XXII. 

The  tides. — Definitions  relating  to  tides. — Causes  of  tides  and 
the  daily  inequality  of  tides. — Effect  of  the  sun. — Priming 
and  lagging. — Luni-tidal  intervals. — Establishment  of  the 
port. — General  laws  of  tides. — Tidal  currents. — Times  of 
high  and  low  water  and  current  data  from  the  Tide 
Tables. — High  or  low  water  by  computation 670-680 

CHAPTER  XXIII. 

Distinction  between  planets  and  fixed  stars. — Distinction  of 
the  principal  planets. — Grouping  and  classification  of 
stars. — List  of  the  navigational  stars. — Constellations  of 
reference. — Stars  referred  to  the  "  Dipper  "  (Ursa  Major), 
to  Orion,  to  the  Southern  Cross. — Identification  in  cloudy 
weather 680-696 

CHAPTER  XXIV. 

General  observations  as  to  the  compasses,  the  sextant,  the 
chronometers,  and  the  charts. — General  duties  of  a  navi- 
gator before  going  to  sea  or  entering  pilot  waters. — 
Discrepancy  in  a.  m.  and  p.  m.  sights  when  abnormal 
refraction  exists. — Error  of  a  ship's  position. — Coefficient 
of  safety. — Advisability  of  keeping  landmarks  in  sight 
when  possible. — General  duties  of  navigator  going  in  or 
out  of  port. — Using  the  seconds  of  data 696-708 

Tables  and  extracts  from  Nautical  Almanac 708-738 

APPENDICES. 

Appendix  A.     Description  of  submarine-bell  system 747 

Appendix  B.  First  compensation  of  a  compass  before  pro- 
ceeding to  sea  and  procedure  in  special  cases  when  com- 
pensating the  compass,  on  one  heading 748 

Appendix  C.  Use  of  azimuth  tables  in  finding  Z,  M,  t,  and  a 
great  circle  course 756 

Appendix  D.  Description  .of  Dr.  Pesci's  nomogram  and  its 
use  to  the  navigator 759 

Appendix  E.  Table  of  compass  points  and  degrees  from  N. 

(to  the  right) 764 


xvi  CONTENTS 

PLATES- 

Plates  I  and  II.  Illuminated  dial  pelorus 53 

Plate  III.  Representation  of  the  nine  soft  iron  rods 113 

Plates  IV  and  V.  Mercator  charts  with  mercator  and  great 

circle  tracks  between  two  given  places 269  and  274 

Plate  VI.  The  principal  stars  around  the  North  celestial 

pole,  d  >  30°    684 

Plate  VII.     The  principal  stars  of  declination  less  than  30° 

and  R.  A.  Oh  to  XIP 688 

Plate  VIII.     The  principal  stars  of  declination  less  than  30° 

and  R.  A.  XIP  to  XXIVh 689 

Plate  IX.     The   principal   stars   around    the   South   celestial 

pole,  d  >  30°    691 

Plate  X.     Conventional    signs    and    symbols,    U.    S.    Hydro- 
graphic  Office  charts 739 

Plate  XI.     Hydrographic  signs,  U.  S.  C.  and  G.  Survey  charts.  740 
Plates  XII  and   XIII.     Topographic  signs,  U.    S.   C.   and  G. 

Survey  charts  741  and  742 

Plate  XIV.     General  abbreviations  on  charts 743 

Plate  XV.     Circles  of  equal  altitude  on  the  mercator  chart..  744 
Plate  XVI.     The  variation  of  the  compass  for  1910 745 


PART  I. 

NAVIGATION  AND  THEORY  OP  THE 
DEVIATIONS  OF  THE  COMPASS. 


NAVIGATION,  THEORY  OF  THE  DEVIATIONS 

OF  THE  COMPASS,  AND  NAUTICAL 

ASTRONOMY. 


CHAPTEE  I. 
DEFINITIONS  AND  GENERAL  OBSERVATIONS. 

Article  1.  Navigation  is  the  science  of  determining  the  posi- 
tion of  a  ship  at  sea,  and  of  .conducting  a  ship  from  one 
position  on  the  earth  to  another. 

2.  There  are  three  general  methods  of  locating  a  ship: 
(1)  When  near  the  coast  by  bearings,  or  bearings  and  dis- 
tances, of  known  objects  on  charts  constructed  by  various 
methods  or  projections  to  represent  the  earth's  surface;  (2) 
by  course  and  distance  made  good  from  a  known  position, 
involving  the  principles  of  plane  trigonometry;  (3)  by  obser- 
vations of  heavenly  bodies,  involving  the  principles  of  spher- 
ical trigonometry.    The  first  may  be  called  pilotage,  the  second 
dead  reckoning,  the  third  nautical  astronomy — all  independ- 
ent in  theory,  but  all  used  practically  in  the  course  of  a  voyage 
from  one  port  to  another  distant  port. 

3.  As  a  ship  is  located  by  the  latitude  and  longitude  of  her 
position,   it  is   proper  to  begin  here  with  the  elementary 
geographical  definitions. 

The  earth  is  an  ellipsoid  of  revolution,  the  equatorial  radius 
being  3963.307  miles,  the  polar  radius  3949.871  miles.  Hence 
the  meridians  are  ellipses,  though  the  parallels  of  latitude  are 
circles.  For  the  general  purposes  of  navigation  the  earth  is 
assumed  to  be  a  sphere. 


2  NAVIGATION 

The  axis  of  the  earth  is  that  diameter  passing  through  the 
poles  of  the  earth  and  about  which  the  earth  daily  revolves 
from  west  to  east. 

The  earth's  equator  is  a  great  circle  of  the  earth  whose 
plane  is  perpendicular  to  the  axis  at  its  middle  point.  The 
plane  of  the  equator  divides  the  earth  into  two  hemispheres, 
the  one  containing  the  north  pole  being  called  the  northern 
hemisphere,  the  one  containing  the  south  pole  being  called 
the  southern  hemisphere.  Every  point  of  the  equator  is  equi- 
distant from  the  poles. 

Terrestrial  meridians  are  great  circles  of  the  earth  passing 
through  the  poles. 

The  meridian  of  a  place  on  the  earth  is  that  meridian 
passing  through  the  place. 

The  prime  meridian  is  that  meridian  from  which  the  longi- 
tude of  places  on  the  earth  is  measured.  The  meridian  of 
Greenwich  is  almost  universally  accepted  as  the  prime 
meridian. 

Parallels  of  latitude  are  small  circles  of  the  earth  whose 
planes  are  perpendicular  to  the  axis. 

The  latitude  of  any  place  on  the  earth's  surface  is  its  angu- 
lar distance  from  the  plane  of  the  equator  north  or  south, 
measured  from  0°  to  90°,  on  the  meridian  passing  through 
the  place. 

The  middle  latitude  of  two  places  in  the  same  hemisphere 
is  half  the  sum  of  their  latitudes.  The  term  is  not  strictly 
applicable  where  the  places  are  situated  on  opposite  sides  of 
the  equator. 

The  longitude  of  any  place  is  the  inclination  of  its  merid- 
ian to  the  meridian  of  some  fixed  station  known  as  the  prime 
meridian,  and  is  measured  by  the  arc  of  the  equator  included 
between  these  two  meridians.  Longitude  is  usually  reckoned 
from  0°  to  180°  east  or  west  of  the  prime  meridian  (usually 
that  of  Greenwich).  It  is  thus  apparent  that  any  point 


DEFINITIONS  AND  GENERAL  OBSERVATIONS  3 

whose  latitude  and  longitude  are  known  can  be  located  on  the 
globe  or  chart  representing  the  earth's  surface. 

The  difference  of  latitude  of  any  two  places  is  the  portion 
of  a  meridian  included  between  the  two  parallels  of  latitude 
passing  through  the  places.  When  both  places  are  on  the 
same  side  of  the  equator,  their  difference  of  latitude  is  found 
by  subtracting  the  smaller  from  the  larger  latitude,  and  when 
the  two  places  are  on  opposite  sides  of  the  equator,  the  differ- 
ence of  latitude  is  found  by  adding  the  two  latitudes ;  when  a 
ship  in  any  latitude  sails  towards  the  pole  of  that  hemisphere 
she  increases  her  latitude,  when  she  sails  away  from  the  pole 
she  decreases  her  latitude;  the  difference  of  latitude  being 
called  N.  or  S.  to  indicate  the  direction  of  the  change. 

The  difference  of  longitude  of  any  two  places  is  the  angle 
at  the  pole,  or  the  corresponding  arc  of  the  equator,  between 
the  meridians  passing  through  the  two  places.  When  the  two 
places  are  in  longitudes  of  a  different  name,  their  difference 
of  longitude  is  found  by  taking  the  sum,  or  360°  —  the  sum. 
The  difference  of  longitude  is  called  E.  or  W.  to  denote  the 
direction  of  change.  In  other  words,  in  combining  latitude 
and  difference  of  latitude,  also  longitude  and  difference  of 
longitude,  the  operation  must  be  performed  algebraically,  the 
terms  N.  and  S.  being  considered  as  opposite  signs,  likewise 
the  terms  E.  and  W. 

The  geographical  and  nautical  miles. — The  geographical 
mile  is  the  length  of  a  minute  of  arc  of  the  equator;  the 
nautical  or  sea  mile  is  the  length  of  a  minute  of  arc  of  a 
circle  having  a  radius  equal  to  the  radius  of  the  curvature 
of  the  meridian  in  the  latitude  of  the  place  considered. 

The  meridians  being  ellipses  flattened  at  the  poles,  the 
linear  length  of  1'  of  the  meridian  is  slightly  different  for 
different  latitudes;  is  least  at  the  equator  and  greatest  at 
the  poles,  its  mean  value  of  6080.27  feet  being  taken  as  the 


4  NAVIGATION 

length  of  the  nautical  mile.  For  navigational  purposes,  the 
geographical  and  nautical  miles  may  be  considered  the  same. 

The  rhumb  line  or  loxodromic  curve  is  a  line  on  the  surface 
of  the  earth  which  makes  a  constant  angle  with  each  succes- 
sive meridian. 

If  a  ship  sails  on  a  loxodromic  curve,  the  constant  angle 
made  by  this  line  with  the  meridian  is  called  the  "  true 
course."  For  trigonometric  computations,  the  course  is  meas- 
ured in  degrees  from  North  or  South  toward  East  or  West, 
according  to  the  data;  though,  in  practice,  navigators  consider 
it  as  estimated,  in  both  hemispheres,  from  the  North  point, 
around  to  the  right,  from  0°  to  360°. 

The  distance  between  two  places,  or  the  distance  run  by  the 
ship  on  a  course,  is  the  length  of  the  loxodrome  joining  the 
two  places  measured  in  nautical  miles. 

4.  Sailing  a  certain  distance  on  a  given  true  course,  the 
distance  North  or  South  from  the  place  left,  measured  on  a 
meridian,  is  the  difference  of  latitude,  and  the  distance  East 
or  West  on  a  parallel  is  the  departure  for  that  latitude,  both 
expressed  in  sea  miles.     Should  the  course  be  due  East  or 
West  on  the  equator,  the  distance  would  be  difference  of  lon- 
gitude. 

Later  on  the  relation  between  departure  and  difference  of 
longitude  will  be  shown  to  be  that  departure  equals  the  differ- 
ence of  longitude  multiplied  by  the  cosine  of  the  latitude. 

5.  The  hearing  of  an  object  or  place  is  the  angle  which  the 
great  circle  passing  through  the  object  (or  place)  and  observer 
makes  with  the  meridian.    It  may  be  expressed  as  true,  mag- 
netic,  or  per  compass,   according   as  the  meridian  is   true, 
magnetic,  or  per  compass.     Bearings,  like  courses,   are  ex- 
pressed practically  by  modern  navigators,   from   the   North 
point,  around  to  the  right,  from  0°  to  360°. 


CHAPTEE  II. 
NAVIGATIONAL  INSTRUMENTS. 

Description   and  Use   of  Logs,   Leads,   Sounding  Machines, 
Charts  and  Protractors. 

6.  Besides  being  provided  with  the  usual  book  outfit  con- 
sisting of  a  log  book,  a  treatise  on  navigation,  one  on  deviation 
of  the  compass,  useful  tables,  azimuth  tables,  nautical  almanac 
for  the  current  year,  also  tide  and  sunset  tables,  corrected 
buoy  lists,  light  lists  and  sailing  directions,  a  file  up  to  date 
of  notices  to  mariners,  and  an  outfit  of  charts  for  the  regions 
to  be  sailed  over,  a  navigator  must  be  provided  with  a  com- 
pass,   azimuth   circle,   pelorus,    sextant,    protractor,    parallel 
rulers,   dividers,   chronometers,   artificial   horizon,   mercurial 
barometer,  a  wet  and  dry  bulb  thermometer,  a  log  and  line 
(preferably  a  patent  log),  hand  and  deep-sea  leads  and  lines, 
and,  if  possible,  Sir  Wm.  Thompson's  or  Captain  Tanner's 
sounding  machine,  a  good  binocular  and  long  glass. 

The  more  important  of  the  instruments  used  in  applying 
the  principles  of  navigation  will  be  considered  in  this  and 
the  succeeding  chapter;  whilst  the  sextant  and  chronometer, 
belonging  properly  to  the  subject  of  nautical  astronomy,  will 
be  considered  in  Part  II. 

7.  Speed  measurers. — The  distance  traversed  by  a  ship  on 
any  course  being  dependent  on  her  speed,  the  accurate  deter- 
mination of  this  speed  is  a  matter  of  great  importance  to  the 
navigator. 

Revolution  of  screw. — The  author  has  found  in  his  expe- 


6  NAVIGATION" 

rience  on  ships  of  various  classes  that  the  revolutions  of  the 
screws  furnish  a  most  convenient  and  accurate  log.  Having 
made  runs  of  known  distances,  in  given  times,  under  favor- 
able conditions,  the  speed  being  uninfluenced  by  currents,  and 
revolutions  carefully  noted,  it  is  easy  to  find  the  coefficient  of 
revolutions  per  minute  for  one  knot  and  to  tabulate  the  revo- 
lutions per  minute  to  make  any  desired  speed.  A  little  ex- 
perience should  teach  the  navigator  what  allowance  to  make 
for  adverse  winds  and  seas,  and  for  any  unusual  trim  of  the 
ship. 

Patent  logs. — There  are  many  mechanical  contrivances,  of 
as  many  various  forms  but  embodying  the  same  general  prin- 
ciples, called  patent  logs,  which,  under  normal  conditions,  are 
very  fair  registers  of  speed.  However,  they  are  far  from  ac- 
curate and  need  careful  watching,  even  when  in  good  working 
order.  If  correct  at  one  speed  they  are  not  liable  to  be  so  at 
a  faster  or  slower  speed,  and  they  register  differently  in  a 
head  or  following  sea.  The  error  of  each  patent  log  should 
be  ascertained  under  varying  conditions  of  wind  and  sea,  at 
different  speeds  and  draft  for  every  run  between  well-deter- 
mined points,  provided  the  speed  is  not  affected  by  tide ;  each 
rotator  and  register  should  be  lettered  and  a  record  kept  of 
their  errors. 

8.  General  description, — The  most  successful  type  may  be 
said  to  consist  of  ( 1 )  the  rotator,  a  hollow  but  enclosed  coni- 
cal shaped  piece  of  brass  with  small  vanes,  towed  astern  by  a 
specially  made  line,  and  caused  to  revolve  more  or  less  rapidly 
according  to  the  ship's  speed,  by  the  pressure  of  water;  (2) 
the  register,  located  on  the  rail  aft,  in  which  cyclometer  gear 
is  worked  by  the  rotator  through  the  agency  of  the  line,  and 
the  miles  and  tenths  of  a  mile  run  thus  indicated  on  the  dial 
plate;  (3)  the  specially  made  line,  the  length  of  which  is  an 
important  factor  in  correct  registering;  experience  can  best 
decide  this  length  for  different  speeds;  a  high  speed  requires 


NAVIGATIONAL  INSTRUMENTS  7 

a  greater  length  than  a  low  speed ;  under  ordinary  conditions 
it  is  advisable  to  use  the  length  of  line  issued  with  the  log 
by  the  manufacturer,  about  400  feet. 


FIG.  1. — Negus  Taffrail  Log. 

It  is  well  to  have  two  patent  logs,  each  of  a  different  manu- 
facture, one  on  each  quarter,  and  the  error  of  each  should  be 
carefully  determined  by  readings  taken  at  times  when  the 
position  of  the  ship  has  been  accurately  found,  and  on  runs 
unaffected  by  currents. 


8  NAVIGATION 

The  Bliss  and  Negus  logs  are  perhaps  the  most  reliable 
ones  on  the  market.  They  are  shown  in  Figs.  1,  2,  and  3. 

The  mechanism  of  the  patent  log  requires  care  and  frequent 
oiling.  In  use,  the  lines  must  he  watched  to  prevent  being 
fouled  by  each  other,  by  seaweed,  cleaning  rags,  barrels,  or 
debris  carelessly  thrown  overboard. 

Instruments  usually  accompany  patent  logs  for  changing 
the  pitch  of  the  rotator  blades  to  correct  an  error  in  register- 


FIG.  2. — Bliss  Star  Log. 

ing;  but,  if  the  error  is  small,  it  is  better  to  leave  it  uncor- 
rected,  and  apply  it  to  the  record  of  speed. 

9.  Patent  electric  log. — This  log  is  the  same  as  the  or- 
dinary patent  log  except  that  the  gearing  registering  the 
knots  and  tenths  of  a  knot  closes  an  electric  circuit  every  time 
a  tenth  is  turned.     The  circuit  thus  closed  magnetizes  a  sole- 
noid, which  in  turn  attracts  a  bar.     This  bar,  by  means  of 
suitable  levers,  moves  a  train  of  gearing  which  registers  the 
tenths  of  a  knot  and  knots.    This  electrically  controlled  regis- 
ter is  placed  on  the  bridge,  or  in  the  pilot  house,  where  it  can 
be  easily  read  by  the  officer  of  the  deck. 

10.  Log  chip,  line  and  sand  glass. — The  speed  of  sailing 

NOTE. — The  Nicholson  Ship  Log  includes  a  clock,  speed  dial,  counter,  record  drum 
and  chart.  It  is  operated  by  floats  in  the  load  level  and  speed  pipes,  the  height  of 
water  in  speed  pipe  depending  on  the  vessel's  speed  ahead.  It  shows  the  speed  and 
mileage  sailed,  and  records  the  speed  on  a  chart  for  every  minute  of  run. 


NAVIGATIONAL  INSTRUMENTS 


ships,  before  the  patent  log  came  into  vogue,  was  determined 
by  the  use  of  the  log  chip,  log  line,  and  sand  glass.  The  log 
chip  was  a  wooden  quadrant  about  5  inches  in  diameter, 
weighted  with  lead  on  the  circular  edge  to  make  it  float  up- 
right, joined  by  a  three-legged  bridle  to  the  log  line  wound 
on  the  reel.  The  two  legs  of  the  bridle  to  the  lower  corners 
were  joined  to  a  pin  which  fitted  into  a  socket  secured  to  the 

leg  attached  to  the  upper  corner. 
The  first  15  or  20  fathoms, 
called  the  stray  line,  was  indicated 
by  a  piece  of  red  bunting,  and  as 
this  bunting  went  over  the  rail 
(the  chip  being  well  clear  of  the 
ship),  the  sand  glass  was  turned 
at  the  order  "  Turn/'  and  at  the 
order  "  Up "  the  line  was  held, 
and  by  a  sharp  jerk  the  chip  was 
untpggled.  The  order  "  Up  "  was 
given  when  the  sand  had  run 
through,  the  length  of  line  out  at 
that  instant,  indicated  by  knots 
and  tenths  of  a  knot  (marks  on  the  line),  gave  the  speed  of 
the  ship. 

The  line  was  subdivided  into  lengths  of  47  feet  and  3  inches 
called  knots,  and  marked  by  short  pieces  of  fish  line,  thrust 
through  the  strands,  and  having  one,  two,  or  three  knots,  etc., 
tied  in  them  according  to  the  number  of  lengths  from  the 
stray  line  mark.  Each  knot  was  subdivided  by  pieces  of  white 
rag  into  lengths  of  two-tenths  of  a  knot  each.  In  marking 
the  line,  the  distance  between  knots  was  gotten  from  the  pro- 
portion, "  length  of  knot  in  feet  is  to  one  sea  mile  in  feet  as 
28  seconds  are  to  the  number  of  seconds  in  an  hour." 

The  glass  itself,  being  a  28-seconds  glass,  was  for  speeds  of 
four  knots  and  under;  for  higher  speeds  a  14-seconds  glass 


FIG.  3. — Bliss  TaffrailLog. 


10  NAVIGATION 

was  used,  and  to  get  the  ship's  speed  per  hour,  using  this 
glass,  the  knots  and  tenths,  as  shown  by  the  line  run  out,  were 
doubled.  Much  depended  on  the  manner  of  heaving  the 
log,  and  errors  of  line  and  glass  were  hard  to  guard  against. 
It  did  not  afford  a  continuous  record  of  the  speed,  and  was 
not  to  be  depended  on  for  speeds  over  10  knots. 

11.  Sounding  has  for  its  object  the  measuring  of  the  depth 
of  water  and  ascertainment  of  the  character  of  the  bottom; 
the  former  being  shown  on  charts  in  fathoms  or  feet,  accord- 
ing to  the  depth,  and  the  latter  noted,  where  known,  as  mud, 
sand,  ooze,  coral,  etc. 

12.  Sounding  apparatus. — The   depth  of  water  is  ascer- 
tained  by   the   sounding   machine   or   the   lead.     There   are 
several  kinds  of  leads  used,  according  to  the  depth  of  water. 

Hand  lead. — On  entering  or  leaving  port  and  in  shallow 
water,  generally  speaking  in  less  than  20  fathoms,  casts  are 
taken  by  the  hand  lead,  a  cylindrical  lead,  weighing  from  7 
to  14  pounds,  attached  to  a  line  of  from  20  to  30  fathoms  in 
length,  properly  marked,  and  that  too  when  wet. 

The  coasting  and  deep-sea  leads. — The  coasting  lead, 
weighing  from  25  to  50  pounds,  is  used  in  depths  from  25  to 
100  fathoms,  beyond  which  depth  the  deep-sea  lead,  weighing 
from  80  to  100  pounds,  becomes  necessary.  The  coasting  and 
deep-sea  leads  are  hollowed  out  at  the  bottom  to  receive  an 
arming  of  tallow  to  bring  up  specimens  of  the  bottom.  The 
lines  are  marked  at  10  fathoms  with  one  knot,  at  20  fathoms 
with  two  knots,  and  so  on,  and  at  every  intermediate  five 
fathoms  with  small  strands;  at  100  fathoms  it  is  marked 
with  a  piece  of  red  bunting. 

It  is  necessary  to  reduce  the  speed  of  the  ship  when  getting 
casts  with  either  the  coasting  or  deep-sea  lead,  and  a  loss  of 
time  ensues;  however,  those  disadvantages  are  obviated  if 
the  ship  is  supplied,  as  every  ship  should  be,  with  a  sounding 
machine,  of  which  class  of  instruments  Sir  Wm.  Thompson's 


NAVIGATIONAL  INSTRUMENTS 


11 


and  Commander  Tanner's  are  certainly  the  best.  Very  accu- 
rate results  can  be  gotten,  without  loss  of  time  or  reduction  of 
speed,  in  depths  up  to  100  fathoms  of  water  with  these  ma- 
chines. 

13.  Sir  Wm.  Thompson's 
sounding  machine  consists 
of  a  wooden  or  metal  frame 
bolted  to  the  deck,  and 
carrying  a  drum  of  about 
one  foot  in  diameter,  on 
which  is  wound  about  300 
fathoms  of  seven-stranded 
flexible  galvanized  steel 
wire  rope;  the  drum  is  con- 
trolled by  a  brake,  and  is 
provided  with  handles  for 
heaving  in,  and  a  dial  plate 
to  record  the  amount  of 
wire  run  out.  To  the  wire 
is  attached  about  9  feet  of 
log  line,  and  an  elongated 
sinker;  between  the  wire 
and  sinker,  fast  to  the  line, 
is  a  small  copper  tube  closed 
by  a  cap  with  a  bayonet 
joint  at  the  top  though 
perforated  at  the  lower  end.  This  tube  is  fitted  to  carry,  when 
sounding,  a  glass  tube  hermetically  closed  at  its  upper  end 
but  open  at  its  lower-  end.  The  glass  tube  is  coated 
on  the  inside  with  chromate  of  silver.  As  the  lead  sinks, 
the  sea  water  is  forced  up  the  tube  in  obedience  to  well- 
known  physical  laws,  and  chemical  action  of  the  salt 
water  changes  the  coating  into  chloride  of  silver  and 
its  color  from  light  salmon  to  a  milky  white,  and  this 


FIG.  4a. 


121  NAVIGATION 

change  takes  place  as  far  up  as  the  water  ascends  in  the  tube. 
A  graduated  boxwood  scale  to  which  the  glass  tube 
is  applied  shows  the  depth  to  which  it  descended, 
and  the  arming  on  the  sinker  shows  the  character 
of  the  bottom.  In  heaving  in,  after  sounding,  care 
must  be  taken  to  keep  the  glass  tube  upright  and 
prevent  water  from  running  into  the  upper  part 
of  the  tube.  Chemically  coated  tubes  which  have 
been  used,  in  shallow  water  may  be  used  again  in 
water  known  to  be  deeper,  and  hence  tubes  dis- 
colored for  only  a  fraction  of  their  length  should 
be  saved  for  future  use. 

The  depth  recorder. — Instead  of  using  the  glass 
tubes  the  depth  recorder  may  be  used.  This  is  a 
metallic  tube  in  which  a  piston  is  forced  up  by 
water  pressure  against  the  tension  of  a  spring;  as 
the  sinker  descends  the  piston  is  forced  up,  carry- 
ing with  it  a  small  marker;  on  being  hauled  up 
after  sounding  the  piston  descends,  but  the  marker 
remains  and  indicates  on  a  graduated  scale  the 
depth  to  which  the  recorder  had  been.  The  marker 
must  be  set  at  zero  and  the  valve  screwed  up  just 
before  use ;  after  each  cast  unscrew  the  nut,  slacken 
the  valve,  and  turn  the  recorder  upside  down  to 
drain  out  the  tube.  Fig.  4b. 

Directions  for  use. — A  pamphlet  containing  full 
directions  for  using  the  machine  is  issued  by 
Messrs.  John  Bliss  &  Co.,  of  New  York,  the  Ameri- 
can agents,  and  accompanies  each  machine. 

The  Thompson  machine  is  shown  in  Fig.  4a. 

14.  Error  of  machine. — The  sounding  machine 

FlQ  4b is  not  reliable  within  the  field  of  action  of  the  hand 

Depth      lead,  and,  always  after  a  short  use,  has  an  error 

Recorder.  ^^  ^  navigator  g^ould  determine.     This  can 

be  easily  done  when  a  ship  is  stationary  in  fairly  deep  water 


NAVIGATIONAL  INSTRUMENTS 


13 


I 


FIG.  5. 


14  NAVIGATION 

by  comparing  depths  gotten  by  the  machine  and  by  use  of  the 
coasting  lead. 

15.  The  Tanner  machine  consists  of  a  metal  frame  in 
three  parts,  a  column  of  steel  surmounted  by  two  brass  discs 
joined  at  their  peripheries.  The  discs  carry  a  shaft,  a  drum 
with  V-shaped  flanges  on  which  is  wound  the  sounding  wire, 
cranks,  compressor  arms,  brake  lever,  register,  and  correction 
table,  at  the  same  time  forming  a  guard  to  prevent  slack  turns 
from  flying  off  the  drum. 

The  wire  consists  of  300  fathoms  of  7-stranded  flexible  gal- 
vanized steel  wire  rope. 

The  brake  mechanism  is  simple  and  direct  acting,  and 
being  in  full  view  of  the  operator  is  easily'  controlled. 
The  drum  is  held  by  moving  the  lever  in  one  direction,  and 
released  by  the  reverse  movement. 

The  cranks  are  not  to  be  unshipped  from  the  machine.  They 
are  provided  with  automatic  locking  bo*lts  which  act  when  pre- 
paring for  action ;  when  these  bolts  are  withdrawn,  the  cranks 
fall  down  each  side  of  the  column,  and  the  handles  are  thrust 
into  the  friction  scores  where  they  are  securely  held,  and  at 
the  same  time  exert  a  slight  friction  on  the  shaft,  almost 
counteracting  the  inertia  of  the  drum  while  sounding. 

The  register,  which  shows  approximately  the  amount  of 
wire  out,  is  directly  beneath  the  shaft,  on  the  port  side  of  the 
machine  as  it  is  set  up  on  the  deck.  The  correction  table  at- 
tached to  the  top  of  the  machine  shows  the  number  of  fathoms 
out  corresponding  to  the  dial  register. 

Experience  has  shown  that  at  10  knots  speed  with  a  depth 
of  more  than  50  fathoms,  the  ratio  of  wire  out  to  actual  depth 
is  about  two  to  one;  at  depths  greater  than  50  fathoms,  and 
with  speed  increasing  to  15  knots,  the  proportion  is  three  to 
one;  and  in  a  heavy  sea  from  four  to  one. 

The  sinker,  which  weighs  about  18  pounds,  has  the  appear- 
ance of  an  ordinary  coasting  lead  with  an  iron  rod  projecting 


NAVIGATIONAL  INSTRUMENTS  15  ' 

from  its  upper  end.  The  sounding  wire  secures  to  an  eye 
in  the  end  of  the  rod.  However,  a  length  of  stray  line  made 
of  cotton  cod  line  or  signal  halliard  stuff,  long  enough  to 
reach  the  machine  when  the  sinker  is  up,  is  often  preferred 
on  account  of  its  flexibility.  The  glass  tube,  which  may  be 
either  the  Tanner-Blish  or  Thompson  tube,  when  in  use  with 
this  machine,  is  carried  in  a  brass  shield  seized  to  the  wire  or 
stray  line  above  the  sinker. 

The  Tanner  combination  lead  is  sometimes  used  instead  of 
the  sinker;  it  weighs  30  pounds,  carries  a  shield  and  sounding 
tube  within  a  central  tube  in  the  lead  itself.  On  account  of 
the  delicate  sounding  tube,  care  must  be  taken  not  to  let  the 
lead  strike  the  ship's  side  when  reeling  in. 

16.  The  Tanner-Blish  tube. — Commanders  Tanner  and 
Blish,  U.  S.  N".  have  patented  a  tube  designed  to  be  used 
continuously  if  not  accidentally  broken.  It  is  a  glass  tube  24 
inches  long  of  small  uniform  bore,  the  walls  of  which  are 
ground  or  clouded.  They  are  translucent  when  dry,  but  clear 
when  wet,  and  the  line  of  demarcation  between  the  clear  and 
the  translucent  part  is  used  to  determine  the  depth  in  the 
same  manner  as  with  the  chemical  tube.  The  bore  is  ground 
spirally  to  counteract  the  effect  of  capillary  attraction.  In 
the  act  of  sounding,  the  upper  end  of  the  tube  being  closed  by 
a  rubber  cap  and  the  lower  end  remaining  open,  the  column 
of  air  will  be  compressed  by  the  water  which  will  enter  it  in 
proportion  to  the  depth  to  which  the  tube  descends,  the  divid- 
ing line  between  the  clear  and  translucent  glass  indicating  the 
height  to  which  the  water  entered  the  tube.  One  end  of  the 
tube  is  left  unground,  and  the  rubber  cap  should  be  placed 
always  on  the  other  end. 

When  its  interior  is  dry  the  tube  will  indicate  the  depth 
without  fail ;  if  not  dry,  no  results  will  follow  the  cast.  Free 
circulation  of  air  is  essential  in  drying  the  bore,  and  to  this 
end  the  cap  must  be  removed.  In  rainy  or  foggy  weather  the 


16  NAVIGATION 

tube  may  be  dried  in  the  engine  room.  The  bore  may  be 
cleaned  at  any  time  by  allowing  a  few  drops  of  alcohol  to  run 
through  it,  back  and  forth,  several  times,  and  then  rinsing  the 
bore  in  fresh  water.  The  tube  should  not  be  used  more  than 
a  dozen  times  without  being  rinsed  in  fresh  water. 

Before  sounding  see  the  tube  dry  and  translucent :  If  any 
part  of  it  is  clear,  put  that  end  to  the  lips  and  draw  dry  air 
through  it  with  long  inhalations  filling  the  lungs,  repeating 
the  process  until  the  whole  glass  is  translucent;  then  put  on 
the  cap  and  proceed  with  the  sounding. 

Among  the  articles  furnished  with  the  machine  is  a  small 
air  pump  for  drying  the  tubes. 

17.  The  machine  in  action. — The  quartermaster  of  the 
watch  and  two  men  are  required  for  the  efficient  operation  of 
the  machine.  The  sounding  tube  is  inserted  in  the  shield, 
open  end  down,  and  the  shield  is  seized  to  the  wire  or  stray 
line;  the  sinker  is  armed,  bent  to  the  wire  or  stray  line,  and 
lowered  over  the  stern,  the  wire  or  line  passing  over  the  roller 
of  the  stern  leader. 

At  the  machine  the  slack  turns  are  reeled  in  by  one  man, 
brake  applied,  and  cranks  thrown  out  of  action,  the  handles 
thrust  into  friction  scores,  and  the  pointer  noted  at  zero. 

When  ready,  the  quartermaster,  with  hand  on  the  brake, 
eases  down  the  sinker  till  it  is  near  the  water,  then  allows  the 
wire  to  run  freely  till  the  sinker  reaches  the  bottom,  or  the 
designated  amount  of  wire  has  run  out;  as  the  wire  runs  out 
it  is  checked  only  as  required  to  prevent  slack  turns,  and  a 
bent  metal  rod  or  hardwood  stick,  called  the  finger  pin,  is 
pressed  lightly  on  the  running  wire,  and  indicates  that  the 
sinker  has  reached  bottom  by  suddenly  approaching  the  deck 
as  the  wire  momentarily  slackens;  at  this  moment  the  brake 
is  applied,  both  cranks  thrown  into  action,  and  the  wire  reeled 
in  by  the  two  assistants.  The  quartermaster,  watching  over 
the  stern,  regulates  the  speed  of  reeling  in,  and  signals 


NAVIGATIONAL  INSTRUMENTS  17 

4 '  stop"  when  the  sinker  is  up,  hauls  it  on  board,  examines 
the  arming,  notes  the  character  of  the  bottom,  the  brakes 
having  been  applied,  and  handles  thrown  out  of  action.  The 
sounding  tube  is  removed  from  the  shield,  applied  to  the  scale 
and  depth  read,  after  which  the  arming  of  the  lead  is  renewed 
and  the  tube  dried  preparatory  to  another  cast. 

If  the  Tanner  lead  is  used  instead  of  a  sinker,  the  shield 
(with  tube)  is  inserted  in  the  central  tube  of  the  lead  before 
or  after  it  is  suspended  over  the  stern.  The  Tanner  machine 
is  shown  in  Fig.  5. 

18.  Use  of  sounding  data. — It  is  often  possible  to  ascertain, 
and  frequently  possible  to  verify,  a  ship's  position  by  sound- 
ings. 

In  thick  and  foggy  weather,  soundings  may  prove  the  only 
safeguard.  The  U.  S.  Naval  Eegulations  are  very  stringent 
in  their  requirements  of  a  navigator  on  this  subject.  How- 
ever, great  care  must  be  exercised  in  trusting  to  soundings 
alone.  Several  taken  at  random  will  seldom  locate  a  ship; 
in  fact  may,  by  misleading,  invite  disaster. 

When  approaching  land  and  on  soundings,  with  no  known 
marks  in  sight  from  which  the  position  of  the  ship  can  be 
gotten,  keep  the  lead  going;  note  the  nature  of  the  bottom  as 
evidenced  by  the  arming  of  the  lead,  the  depth  and  time  of 
each  sounding,  and  the  course  and  distance  to  the  next  one. 
Take  a  piece  of  tracing  paper  or  muslin  sufficiently  large  to 
cover  the  area  likely  to  include  the  ship's  position  during  the 
runs  considered;  rule  it  with  the  meridians  of  the  chart  in 
use.  Mark  an  assumed  position  in  such  a  part  of  the  tracing 
as  to  have  sufficient  room  for  the  courses  and  distances  from 
that  position,  write  near  it  the  depth  and  character  of  bottom 
at  the  first  sounding.  Lay  down  to  the  scale  of  chart  the 
course  and  distance  from  the  first  position  to  that  of  the  sec- 
ond cast,  note  as  before  the  depth  and  character  of  bottom  at 
that  cast.  Having  a  traverse  of  several  positions  and  sound- 


18  NAVIGATION 

ings,  move  the  tracing  up  and  down,  and  from  side  to  side, 
keeping  the  ruled  meridians  parallel  to  those  of  the  chart, 
until  the  soundings  and  character  of  bottom  on  the  tracing 
correspond  in  close  agreement  with  those  on  the  chart.  In 
this  way  a  fair  location  of  the  ship  on  the  chart  may  be  got- 
ten, even  though  exact  correspondence  of  data  is  not  found. 

19.  Charts  are  representations  of  certain  parts  of  the  earth's 
surface  upon  a  plane  surface,  in  accordance  with  some  one  of 
several  definite  systems  of  projection,  an  effort  being  made  to 
satisfy  the  conditions  that  any  two  distances  from  the  center 
of  the  chart  shall  have  the  same  ratio  as  the  corresponding 
distances  on  the  earth's  surface,  that  the  ratio  of  the  areas  of 
definite  limits  on  the  chart  and  the  ratio  of  the  same  limits  on 
the  earth  shall  be  the  same,  and  that  the  ratio  of  all  cor- 
responding angles  shall  be  unity.  However,  since  the  earth 
is  an  ellipsoid,  and  the  representation  of  it  is  on  a  flat  surface, 
i  t  is  evident  there  must  be  distortion,  and  the  effort  should  be 
to  make  this  a  minimum. 

Charts  are  made  primarily  for  the  use  of  seagoing  men,  and 
show  meridians  and  parallels  of  latitude,  the  details  of  the 
coast,  light  houses,  life  saving  stations,  mountains  and  promi- 
nent hills  near  the  coast,  soundings,  dangers  and  shoals,  nature 
of  bottom,  light  ships,  fog  signals,  buoys,  beacons,  tidal  and 
current  data,  variation,  etc.,  all  of  which  may  assist  the  navi- 
gator in  making  a  successful  voyage. 

Charts  are  divided  into  general,  sheet,  and  harbor  charts; 
the  elaborateness  of  detail  depending  on  the  scale,  and  the 
character  depending  on  the  purposes  to  be  served. 

General  charts  comprise  an  entire  ocean,  or  a  large  part  of 
it,  or  a  considerable  extent  of  coast  line  with  adjacent  waters. 
In  addition  to  information  referred  to  above,  general  charts 
should  show  the  principal  sailing  routes. 

A  sheet  chart  is  a  detached  portion  of  a  general  chart  and 
is  made  on  a  larger  scale.  It  usually  gives  the  information 


NAVIGATIONAL  INSTRUMENTS  19 

first  referred  to,  and  enables  a  navigator  to  use  the  channels 
for  entering  the  bays  and  large  harbors. 

A  harbor  chart  is  one  of  a  harbor  or  a  harbor  and  its  ap- 
proaches; the  curvature  of  the  earth  is  not  considered,  but 
owing  to  the  small  extent  included  in  the  chart,  there  is  no 
distortion.  It  is  made  by  assuming  an  observation  spot,  the 
latitude  and  longitude  of  which  are  determined,  measuring  a 
base  line,  cutting  in  signals  by  sextant  or  theodolite,  filling 
in  the  detail,  and  running  cross  lines  of  soundings. 

When  the  approaches  to  a  harbor  are  of  any  extent,  as  for 
the  U.  S.  harbors,  the  charts  made  by  the  Coast  Survey  are 
on  the  polyconic  projection. 

20.  Systems  of  projection. — There  are  three  principal  sys- 
tems of  projection  used  in  chart  making,  (a)  Polyconic,  (b) 
G-nomonic,    (c)    Mercator,   a   so-called  projection  by  which 
rhumb  lines  appear  as  right  lines  on  plane  surfaces. 

21.  The  polyconic  projection. — In  this  the  earth's  surface 
is  developed  on  a  series  of  cones  tangent  to  the  earth,  a  differ- 
ent one  for  each  parallel,  the  parallel  forming  the  base  of  the 
cone,  the  vertex  of  which  is  on  the  axis  of  the  earth  produced. 

The  parallels  of  latitude  are  developed  as  arcs  of  circles, 
but  being  from  different  centers  and  with  different  radii  they 
are  not  parallel;  the  meridians,  except  the  middle  one,  are 
curved  and  converge  toward  the  pole. 

The  degrees  of  latitude  and  longitude  are  projected,  prac- 
tically in  their  true  length,  in  consequence  there  is  no  distor- 
tion at  the  middle  meridian,  and  very  little  anywhere,  if  the 
limits  of  the  chart  in  longitude  are  narrow.  As  the  minutes 
of  latitude  are  practically  of  the  same  length,  one  scale  of 
distances  may  be  used  for  any  part  of  the  chart. 

The  geodesic  line  between  two  places  (the  shortest  distance 
on  the  spheroid)  will  be  projected  practically  as  a  straight 
line  in  its  true  length,  but  the  loxodrome  will  be  projected  as 


20  NAVIGATION 

a  curved  line  and  the  true  course  will  change  from  the  begin- 
ning to  the  end  of  the  voyage.  For  this  reason  practical  sea- 
going people  find  it  an  inconvenient  projection.  Fortunately, 
however,  the  course  may  be  taken  as  a  straight  line  on  a  chart 
of  large  scale. 

Certain  meridians  and  parallels  are  subdivided  in  different 
parts  of  the  chart;  and  whenever  it  is  desired  to  plot  the 
ship's  position,  the  subdivisions  nearest  to  the  position  must 


FIG.  6. 


be  used,  and  the  same  rule  must  be  followed  in  taking  off  the 
latitude  and  longitude  of  a  position. 

22.  Equations  for  the  coordinates  of  a  polyconic  chart. — 
If,  as  in  Fig.  6,  N  is  the  normal  terminating  in  the  minor 
axis,  and  L  the  angle  it  makes  with  the  major  axis;  a,  the 

a, 

equatorial  radius ;  e,  the  eccentricity ;  then  N  = — — 

{]L~~ G  sin    Jj) 

and  r  =  N  cot  L  is  the  slant  height  of  the  tangent  cone 
and  the  radius  of  the  developed  parallel,  the  developed  par- 
allel being  a  circle.  Since  in  practice  it  would  be  incon- 


NAVIGATIONAL  INSTRUMENTS  21 

venient  to  describe  the  arcs  with  radii,  they  can  better  be 
drawn  by  constructing  them  from  their  equations,,  and  it  will 
be  found  convenient  to  have  x  and  yf  the  rectangular  co- 
ordinates of  a  point,  whose  latitude  is  L°  and  whose  longitude 
differs  from  that  of  the  middle  meridian  by  n°,  expressed  as 
functions  of  the  radius  of  the  developed  parallel  and  the 
angle  the  radius  makes  with  the  middle  meridian.  Let  6  be 
this  angle  (Fig.  7),  the  origin  being  taken  at  L,  the  point  of 
intersection  of  any  parallel  with  the  middle  meridian;  the 
middle  meridian  as  axis  of  Y,  the  perpendicular  through  L  as 
axis  of  X,  then  the  coordinates  of  any  point  P  whose  latitude 
is  L°  and  longitude  is  n°  from  the  middle  meridian,  will  be 
x  =  r  sin  0  =  N  cot  L  sin  0  (1) 

y  —  r  (1  —  cos  6}  =  N  cot  L  versin  6  (2) 

where  0  is  some  function  of  n° . 

To  determine  the  relation  of  n  and  0,  it  is  only  necessary  to 
remember  that  the  parallels  are  projected  with  their  true 
length,  in  other  words, "the  distance  LP,  Fig.  7,  equals  the 
distance  between  L  and  P  on  the  spheroid,  measured  on  the 
parallel  passing  through  L  and  P,  therefore  angles  at  the 
centers  of  the  two  arcs  will  be  in  inverse  proportion  to  the 
radii,  or  N  cot  L  X  0  =  N  cos  L  X  n>° ;  therefore, 

e  =  n°  sin  L.  (3) 

These  three  equations  are  sufficient  to  project  any  point  of 
the  spheroid  given  by  its  latitude  and  the  number  of  degrees 
of  longitude  from  the  middle  meridian. 

In  tables  prepared  by  the  U.  S.  Hydrographic  Office,  the 
elements  of  the  terrestrial  spheroid,  and  the  coordinates  of 
curvature,  x  and  y,  are  tabulated  in  meters. 

23.  Construction  of  a  polyconic  chart  (Fig.  8). — Draw  a 
straight  line  LL'  for  the  middle  meridian;  using  Table  IV 
of  the  projection  tables  referred  to,  take  out  for  Lat.  L  from 
column  Dm  (degree  of  Lat.)  the  distance  which  is  laid  off 


22  NAVIGATION 

from  L  according  to  the  scale  of  the  chart.  It  is  equal  to 
Lm^  and  locates  the  parallel  for  Lat.  mx  at  the  middle  me- 
ridian; lay  off  m±m2,  taken  out  of  the  tables  for  Lat  m±,  and 
locate  Lat.  m2?  continue  this  till  the  rectified  arc  of  the  me- 

DIAGRAMMATIC  DRAWING  OF  A  POLYCONIC  CHART. 
Distorted  for  illustration. 


FIG.  8. 

x  and  y  the  coordinates  for  latitude   L°  and  for  n°   of 
longitude  each  side  of  the  middle  meridian. 

ridian  LL'  is  completed.  Through  the  points  thus  found, 
draw  the  perpendiculars  KLK,  Kvm^K^  etc.,  to  represent  the 
axis  of  X  in  each  case. 


NAVIGATIONAL  INSTRUMENTS  23 

On  these  perpendiculars,  set  off  to  E.  and  W.  of  the  middle 
meridian  the  abscissas  x,  and  on  lines  at  right  angles  towards 
the  pole  the  ordinates  y.  The  coordinates  are  taken  from 
Table  II,  or  Table  III,  of  "  The  Projection  Tables/'  accord- 
ing to  the  detail  required,  and  laid  down,  according  to  the 
given  scale,  for  each  parallel  of  latitude  and  each  required 
longitude. 

The  diagrammatic  sketch  of  a  polyconic  chart,  Fig.  8,  will 
serve  to  illustrate  the  distortion  at  meridians  removed  from 
the  middle  one.  The  series  of  cones  divides  the  surface  to  be 
projected  into  a  series  of  zones,  each  zone  tangent  to  those 
adjacent  to  it  above  and  below  'only  at  the  middle  meridian, 
and  separating  from  them  to  the  eastward  and  westward.  To 
complete  the  tangency  and  make  the  chart  continuous,  s'n2, 
s2nly  etc.,  should  be  stretched,  so  that  the  lower  edge  of  the 
zone  n2m2n2  will  coincide  with  the  upper  edge  S2m2s2  of  the 
lower  zone  along  a  middle  curve,  even  then  producing  slight 
distortion  which  would  increase  with  the  longitude  of  points 
from  the  middle  meridian. 

24.  Gnomonic  projection. — In  this  system,  the  earth's  sur- 
face is  projected  by  rays  from  the  center  upon  a  plane  tangent 
to  the  earth's  surface  at  a  given  point,  so  it  is  apparent  that 
all  great  circles  will  be  projected  as  straight  lines.  The 
great  circle  track  is  represented  as  a  straight  line  and  for 
this  reason  such  charts  are  often  called  great  circle  charts. 
Except  when  the  point  of  tangency  is  at  the  pole,  the  parallels 
will  be  conies.  The  U.  S.  Hydrographic  Office  issues  a  series 
of  charts  to  cover  the  various  cruising  grounds  of  the  world, 
and  on  these  are  diagrams  with  full  explanations  for  their  use 
in  finding  the  great  circle  course  and  distance  between  two 
points. 

The  polar  chart. — The  simplest  form  of  the  gnomonic 
chart  is  the  polar  chart,  Fig.  10,  in  which  the  tangent  plane 
is  tangent  at  the  pole;  on  such  a  chart  great  circles  are 


24  NAVIGATION 

straight  lines,  the  meridians  are  right  lines  radiating  from 
the  pole,  whilst  the  parallels  are  projected  as  circles  whose 
center  is  the  pole;  though  accurate  in  high  latitudes,  this 
projection  would  give  a  distorted  chart  for  low  latitudes. 

25.  Construction  of  a  polar  chart. — In  Fig.  9,  let  AB  be 
the  plane  tangent  at  the  pole  8,  pp'  the  parallel  to  be  pro- 
jected, p-ip\  will  be  the  diameter  of  the  projected  parallel 
which  will  be  a  circle.  Let  R  =  earth's  radius,  x  the  radius 


FIG.  9. 

of  projected  parallel;  therefore,  x  =  R  cot  L.  The  radius 
of  the  parallel  of  45°  Lat.,  after  projection,  is  R  since  cot 
45°  is  unity,  and  this  radius  is  called  the  radius  of  the 
chart;  and  to  find  the  radius  of  any  other  parallel  on  the 
chart,  we  have  x  =  R  cot  L,  where  R  is  the  number  of  units 
of  the  scale  in  the  radius  of  the  projected  45th  parallel.  To 
find  the  length  of  a  degree  of  longitude  on  any  parallel  we 

have 


360 


Eeferring  to  Fig.  10,  let  us  construct  a  polar  chart  to  com- 
prise the  earth's  surface  from  45°  N.  Lat.  to  the  pole.  Par- 
allels at  intervals  of  5°,  meridians  at  intervals  of  15°,  R  =  36 


NAVIGATIONAL  INSTRUMENTS 


millimetres;  taking  cotangents  of  latitude  to  nearest  third 
decimal  place,  we  have  the  following  values  of  x\ 

A  POLAR  CHART. 


FIG.  10. 
Scale  :  Radius  lat.  45°=  36  millimetres. 

For  radius  of  50°,  x  —  36  X  -839  =  30.204  millimetres. 

«  55°,  x  =  36  X  .700  =  25.200 

"  60°,  x  =  36  X  .577  =  20.772 

«  65°,  x—  36  X  .466  =  16.776 

«  70°,  x  —  36  X  .364  =  13.104 

"  75°,  x  —  36  X  .268=    9.648 

"  80°,  x—  36  X  .176=    6.336 

With  these  different  radii  and  also  R  =  36  mm.,  draw  the 

concentric  circles,  number  the  parallels  properly,  and  with  a 


26 


NAVIGATION 


ds 


dy 


dm 


protractor  divide  the  outer  circumference  into  15°  divisions, 
drawing  radii  through  the  points.  Mark  one  of  these  me- 
ridians 0°  and  the  others  as  indicated  on  the  chart;  W.  longi- 
tude to  right,  E.  longitude  to  left. 

Use  of  the  polar  chart. — The  navigator,  by  drawing  a 
straight  line  between  the  two  required  points,  can  see  at  a 
glance  whether  it  is  practicable  to  follow  the  great  circle 
route ;  take  off,  if  desirable  so  to  do,  the  latitude  and  longitude 
of  the  vertex,  and  of  other  points  along  the  track,  transfer  them 
to  the  mercator  chart,  and  then  lay  courses  on  the  mercator 
chart  from  point  to  point  of 
this  transferred  track.  For 
instance,  suppose  it  is  de- 
sired to  go  by  great  circle 
route  from  a  point  off  Sable 
Island,  Lat.  45°  K,  Long. 
60°  W.  to  a  point  in  the 
English  Channel,  Lat.  50° 
N".,  Long.  0°.  Use  the 
polar  chart,  Fig.  10.  Draw 
the  straight  line  A B  and 
then  PV  perpendicular  to 

it  from  P.  The  position  of  V  which  is  the  vertex,  the 
point  nearest  the  pole,  is  readily  seen  by  inspection;  or 
measure  the  distance  PV  in  millimetres,  divide  it  by  36,  the 
scale  of  the  chart,  and  the  result  is  the  natural  cotangent 
of  the  Lat.  of  V.  For  the  longitude  measure  with  a  pro- 
tractor the  angle  between  the  meridian  of  V  and  the  one 
adjacent  to  it,  applying  the  angle  properly. 

26.  The  loxodrome. — Before  taking  up  the  subject  of 
the  mercator  chart  and  its  construction,  it  is  desirable  to  con- 
sider mathematically  the  loxodromic  curve  and  to  find  the 
equation  expressive  of  the  difference  of  longitude,  reckoned 
from  the  point  at  which  this  curve  intersects  the  equator  in 


NAVIGATIONAL  INSTRUMENTS  27 

terms  of  any  given  latitude  and  the  constant  course  C.  The 
deduction  following  is  taken  principally  from  Walker's  Navi- 
gation by  permission  of  the  author  of  that  excellent  treatise. 
In  Fig.  11,  let  P  be  any  point  on  the  earth's  surface,  situ- 
ated on  the  meridian  PE  and  on  the  loxodrome  PQ,  and  let 
EE'  be  the  equator.  Denote  the  equatorial  radius  by  a,  the 
radius  of  the  parallel  of  P  by  x,  and  the  radius  of  curvature 
of  the  meridian  PE  at  point  P  by  p.  Denote  the  latitude  of 
P  by  L,  its  longitude  by  A,  the  course  by  (7,  the  earth's  me- 
ridional eccentricity  by  e}  the  longitude  at  which  the  loxo- 
drome crosses  the  equator  by  A0.  If  we  let  y  represent  the 
number  of  sea  miles  along  the  parallel  of  P  included  between 
the  prime  meridian  and  the  meridian  of  P,  and  as  the  longi- 
tude of  P  is  A,  we  shall  have,  when  A  is  in  minutes  of  arc 

representing  sea  miles.  -^  —  —  =    the  circular  measure  of 

x        a 

the  angle  between  the  planes  of  the  two  meridians.  Let  ds 
denote  the  rate  of  the  point  P  along  the  tangent  to  the  loxo- 
drome, and  dm  and  dy  be  its  rectangular  components  in  the 
tangent  plane  at  P.  Then, 


Now,  since  the  element  of  the  terrestrial  meridian  at  its 
intersection  with  any  parallel  of  latitude  is  equal  to  the 
product  of  the  radius  of  curvature  and  element  of  latitude  at 
that  point,  in  accordance  with  the  principle  that  the  radius 
of  curvature  varies  inversely  as  the  angle  between  consecu- 
tive normals,  or,  in  this  case,  as  the  element  of  latitude,  we 

have  dm  =  PdL,  therefore,  tan  C  =  3fL  =  -^L  (5) 

am       apaL 


But  from  differential  calculus  we  have  the  following  well- 
known  expressions  for  the  properties  p,  x,  and  e,  of  the  ter- 
restrial spheroid  considered  as  an  ellipsoid  of  revolution: 


28  NAVIGATION 

a  (1  —  e2)  _         a  cos  L 

~~  (l-e2sin2Z)^         ~  (1  —  ez  sin2 

c  =  0.003407562  the  compression  of  the  earth, 
hence  by  substitution  in  (6), 

ill  -     a  ^an  ^(1  —  g2)  ^  m 

"(I—  e2sin2L)  cos£* 

By  integrating   (7)   between  the  limits  A0  and  A,  0  and  L, 
we  shall  have  A  —  A0,  the  difference  of  longitude  required. 


=  a  tau  g 
but 


r 

o  cos  i(l  —  e2  siu'  Z) 

*    (1  —  «2)cosirfi 

o  coW(l-^ein^)  (8) 

f  [^^^^  -  ,  \L  -^os,LdL  "I         (9) 
LJo     cos2  i  Jo  1  —  0  'sm-'Lj 


cos2  L         Jo  1  — sin2  L 

T       i  i_  1  4-  sinZ 


1  +  sinL      1  —  sin  Z/  3  1  —  smL 

e  cos  L  e  cos  L 


[L  _e^o^LdL_  _  (L,f     ecosL  e  cos  L    \  ^^ 

Jo  1  —  e2sin2^"~Jo     \l  +  esinj6      1  —  e  sin  L) 

=  »  log  J+eBinL 
&  1  —  e  Bin  L 


hence  }  log  =  log  tan          + 

also 

l+eBinX  =  esinZ      g^Z+g'sin'X  + 
5  1  —  e  sin  X  3  5 

whence  by  substitution  in  (9), 

A  _  ;0  =  a  tan  (7  flog  tan  ^  +  -^-] 

-.'(sin  L  +  **£±  +  ^^  +  etc.)]  (10) 
or  if  we  put  e  sin  L  =  sin  <£,  we  shall  have, 


NAVIGATIONAL  INSTRUMENTS  29 


In  this  the  logarithm  is  Naperian,  so  to  reduce  to  common 
logarithms  divide  by  the  modulus  m  =  .434294482  ;  then  in- 
troducing the  value  of  a,  the  equatorial  radius,  3437.74677 
minutes  of  equatorial  arc,  or  nautical  miles,  we  have, 

A  —  A  =  D  =  tan  G  ~  7915.  704  flogu  tec  (^  +  | 

-*logl.  tan 


or  D  =  7915/704  tan  0  logloT= '  (12) 


27.  The  mercator  chart,  —  At  the  equator  on  the  spheroid  a 
degree  of  longitude  equals  a  degree  of  latitude,  but  as  the 
poles  are  approached  the  length  of  a  degree  of  longitude 
becomes  less,  and  finally  zero  at  the  poles,  while  the  degrees 
of  latitude  undergo  but  slight  change.  On  the  mercator 
chart,  which  owes  its  origin  to  one  Gerard  Mercator,  who  lived 
in  Flanders  from  1512  to  1574,  the  meridians  are  drawn 
parallel  to  each  other  and  perpendicular  to  a  straight  line 
representing  the  earth's  equator,  the  distance  apart  on  the 
chart  being  the  distance  between  them  on  the  spheroid,  in 
minutes  of  arc  on  the  equator,  multiplied  by  the  scale  of  the 
chart;  thus  the  departure  on  the  various  parallels  of  latitude 
is  increased  and  made  equal  to  the  difference  of  longitude. 

As  a  compensation,  and  in  order  to  preserve  the  proportion 
that  exists  between  degrees  of  latitude  and  longitude  at  differ- 
ent parts  of  the  earth's  surface,  and  to  maintain  the  relative 
position  and  direction  of  objects  charted,  the  infinitesimal 
divisions  of  a  meridian  in  the  latitude  of  any  parallel  must 
be  increased  in  the  same  ratio  as  the  departure  on  that  par- 
allel. Eegarding  the  earth  as  a  sphere  this  ratio  would  be  as 


30  NAVIGATION 

sec  L  to  1,  though  allowance  is  usually  made  for  the  merid- 
ional eccentricity.  The  series  of  parallels  will,  therefore, 
appear  as  a  series  of  right  lines  parallel  to,  and  at  such  in- 
creasing distance  from  the  equator  as  to  maintain  the  re- 
quired equality  of  angles  and  make  the  loxodromic  curve  a 
straight  line. 

Let  D  denote  the  difference  of  longitude  between  the 
meridians  marking  the  intersections  of  the  loxodrome,  first 
with  the  equator,  and  second  with  any  parallel  of  latitude  L; 
and  let  M  denote  the  augmented  latitude  for  latitude  L  on 
the  chart,  we  then  have, 

D  =  M  tan  C; 

but  this  D  and  this  0  are  the  same  ones  that  appear  in  equa- 
tion (12)  above,  therefore, 


tan 


M=  7915/704  log! 


Ei 


(13) 


The  value  of  M  in  nautical  miles  for  any  latitude  L  is 
known  as  the  meridional  parts  for  that  latitude.  The  merid- 
ional parts  have  been  computed  and  are  found  tabulated  in 
various  works ;  in  Bowditch's  Navigator  and  in  "  The  Useful 
Tables"  they  are  found  in  Table  3  from  0°  to  79°  59',  at 
intervals  of  V .  The  value  of  the  compression  used  in  com- 

Putin8Table3was  29085- 

Since  the  meridional  parts  for  any  latitude  L  are  the  num- 
ber of  nautical  miles,  or  1'  of  longitude,  in  the  meridional 
distance  from  the  equator  to  the  parallel  of  latitude  L  on  the 
Mercator  chart,  the  meridional  difference  of  latitude  for 
Lats.  L  and  L'  is  the  difference  of  meridional  parts  for  those 
latitudes  if  of  the  same  name,  or  the  sum  of  the  meridional 
parts  for  Lats.  L  and  L'  if  of  a  different  name ;  in  other  words, 


NAVIGATIONAL  INSTRUMENTS  31 

it  is  the  algebraic  difference  in  either  case,  represented  by  m, 
or  ra  =  M2  ~  M ^ . 

If  in  equation  (12)  we  regard  e  as  zero,  in  other  words,  if 
the  earth  be  considered  as  a  sphere  instead  of  as  a  spheroid, 
we  shall  have, 

D  =  7915'.704  tan  C  Iog10  tan  f  45°  +  £\  .         (14) 

28.  Construction  of  a  Mercator  chart. — The  method  of 
construction  depends  on  whether  the  chart  is  to  include  the 
equator,  and  if  so,  the  position  of  equator  on  the  chart;  and 
also  whether  the  scale  is  to  depend  on  the  extent  of  paper  in 
the  direction  of  the  meridians,  or  at  right  angles  to  them.  By 
the  term  scale  is  meant  the  actual  length  on  the  chart  of 
1'  of  arc  of  longitude  on  the  earth's  surface.  If  the 
chart  includes  the  equator,  the  values  of  M  as  taken  from 
Table  3  are  to  be  measured  off  directly  from  the  equator  in 
the  proper  direction. 

If  the  chart  does  not  include  the  equator,  then  the  lowest 
parallel  to  be  represented  on  the  chart  is  taken  as  the  origin 
of  parallels,  and  the  distance  from  it  to  any  other  parallel  is 
the  meridional  difference  of  latitude,  as  explained  in  the  pre- 
ceding paragraph. 

If  the  extent  of  paper  between  the  upper  and  lower  parallel 
is  limited,  this  distance  is  measured  and  divided  by  the  merid- 
ional difference  of  latitude  for  the  two  parallels,  and  the 
result  is  the  length  of  V  of  arc  of  longitude  or  the  scale  of  the 
chart.  Multiply  this  by  60  for  the  length  of  one  degree,  and 
the  length  of  one  degree  by  the  number  of  degrees  of  longitude 
to  be  charted,  to  obtain  the  distance  between  the  Eastern  and 
Western  neat  lines  or  bounding  meridians  of  the  chart. 

In  case  the  paper  is  limited  in  an  East  and  West  direction, 
draw  a  line  near  the  bottom  of  the  paper  to  represent  the  low- 
est parallel ;  divide  this  line  into  as  many  equal  parts  as  there 
are  degrees  of  longitude  to  be  represented  on  the  chart ;  then 


32  NAVIGATION 

the  length  of  one  of  these  divisions  divided  by  60  gives  the 
scale  of  the  chart.  This  scale  multiplied  by  the  meridional 
difference  of  latitude  for  the  parallel  representing  the  origin 
and  any  other  parallel  L'  will  give  the  actual  distance  between 
the  two  parallels. 


FIG.  12. 

As  an  illustration,  construct  a  Mercator  chart  to  include 
latitudes  45°  K  to  49°  N.  and  longitudes  141°  W.  to  145°  W., 
scale  14.4  mm.  =  1°  longitude,  and  subdivide  each  degree  of 
both  latitude  and  longitude  into  six  divisions  of  10'  each  (see 
Fig.  12). 


NAVIGATIONAL  INSTRUMENTS 


33 


In  the  center  of  the  paper  draw  a  vertical  line  to  represent 
the  middle  meridian  143°  W.  Near  the  lower  edge  of  the  sheet 
erect  a  perpendicular  to  this  line  which  is  the  southern  inner 
neat  line  of  the  chart,  or  45th  parallel  of  latitude.  From  the 
intersection  of  these  two  lay  off  distances  of  14.4  mm.  and 
28.8  mm.  on  the  parallel,  both  Eastward  and  Westward,  and 
through  these  points  draw  lines  parallel  to  the  middle  me- 
ridian; the  two  outer  lines  will  be  the  extreme  meridional 
neat  lines  of  the  chart,  or  the  meridians  of  141°  W.  for  the 
Eastern  limit,  and  145°  W.  for  the  Western  limit.  It  now 
remains  to  locate  and  draw  in  the  parallels;  their  distances 
from  the  origin  of  parallels  is  determined  from  the  following 
self-explanatory  table : 


Latitude. 

Mer.  Pts., 
M. 

Mer.  Diff.  Lat. 
or  m. 

Multi- 
plier. 

Distance  to 
parallels  in 
Millimetres. 

The  Multi- 
14.4 
plier  is  -^-be- 

45° 

3013.4 

14.4 

cause  1°  =14.4 

46 

3098.7 

85.3 

60 

20.472 

ram.    .*.  1'  = 

47 

3185.6 

172.3 

—  m  24 

41.328 

14.4 

48 

3274.1 

260.7 

62.568 

-™-  mm.      Al- 

49 

3364.4 

351. 

84.240 

ou 

ways  check  the 

work   as    indi- 

cated. 

Check  869. 2    x     .24   =  208.608 

On  the  right  or  Eastern  neat  line  lay  off  the  distances  in  mil- 
limetres, as  shown  in  the  table  above,  from  the  lowest  parallel ; 
20.472  mm.  to  the  46th,  41.328  mm.  to  the  47th,  62.568  mm. 
to  the  48th,  and  84.240  mm.  to  the  49th  parallel.  Through 
the  points  thus  determined,  rule  right  lines  perpendicular  to 
the  meridians  and  these  will  be  the  various  parallels  required. 
Check  the  rectangularity  of  the  construction  by  measuring  the 
diagonals  which  should  be  equal.  Draw  the  outer  neat  lines 
of  the  chart  at  distances  desired,  extend  to  them  the  meridians 
and  parallels.  Subdivide  the  degrees  of  latitude  and  longi- 
tude between  the  inner  and  outer  neat  lines  by  using  proper-  • 
tional  dividers  or  by  a  geometrical  process. 


34  NAVIGATION 

29.  Advantages  and  disadvantages  of  the  different  projec- 
tions.— The  polyconic  chart  has  practically  no  distortion  along 
the  middle  meridian,  is  well  adapted  to  all  latitudes,  shows 
areas  in  their  proper  relation  as  to  magnitudes,  and  permits 
the  use  of  a  single  scale  of  distance  anywhere.    However,  the 
meridians  and  parallels  are  curved,  the  rhumb  line  is  curved, 
and  there  is  distortion  as  the  longitude  departs  from  the  mid- 
dle meridian.     The  gnomonic  chart  is  useful  simply  for  find- 
ing the  great  circle  course  and  distance;  for  navigational  pur- 
poses it  is  useful  in  high  latitudes  where  the  Mercator  projec- 
tion fails.     It  gives  a  distorted  idea  of  the  earth's  surface  at 
points  some  distance  from  point  of  tangency  of  plane  of  pro- 
jection, and  on  it  the  rhumb  line  is  curved.     On  the  special 
form  known  as  the  polar  chart  the  rhumb  line  is  spiral. 

For  navigational  purposes  the  Mercator  chart  is  by  far  the 
most  convenient.  The  shapes  of  small  areas  are  but  little 
distorted;  latitudes  and  longitudes  may  be  laid  down  easily 
and  accurately.  The  ship's  track  is  a  straight  line,  and  the 
angle  this  line  makes  with  any  meridian  is  the  course.  How- 
ever, it  cannot  be  used  in  very  high  latitudes  advantageously, 
the  expansion  being  too  great.  The  relative  areas  of  land  or 
bodies  of  water  in  different  latitudes  cannot  be  compared  by 
the  eye.  The  first  objection  is  obviated  by  using  a  polar 
chart  for  those  regions,  the  second  is  unimportant  to 
mariners. 

30.  Conventional    notation,    and    hydrographic    signs. — 

Soundings  are  in  feet  or  fathoms,  as  indicated  under  the  title, 
and  refer  to  the  plane  of  mean  low  water  for  Atlantic  Coast 
charts,  that  of  the  mean  of  lower  low  water  of  each  tidal  day 
for  Pacific  Coast  charts,  issued  by  U.  S.  Coast  Survey.  On 
British  Admiralty  charts  the  plane  of  reference  is  low  water 
of  ordinary  spring  tides. 

Upon  harbor  and  bay  charts  of  the  United  States,  the  con- 
tour lines,  or  lines  of  equal  depth,  are  traced  for  every  fathom 


NAVIGATIONAL  INSTRUMENTS  35 

up  to  three  fathoms.  Within  the  three-fathom  mark  the  chart 
is  shaded,  the  shading  being  lighter  for  each  fathom;  beyond 
the  three-fathom  line  there  is  no  shading.  On  section  charts 
of  the  coast,  contours  of  10,  20,  30,  and  100  fathoms  are 
shown.  Only  the  latter  curve  is  given  on  large  general  charts. 

No  bottom,  for  instance,  at  50  fathoms,  — —  . 

Nature  of  the  bottom. — The  material  of  the  bottom  is  ex- 
pressed by  capital  letters,  M  for  mud,  G  for  gravel,  S  for 
sand,  etc.;  colors  or  shades  by  two  small  letters,  yl.,  yellow, 
gy.,  gray,  etc. ;  other  qualities  by  three  small  letters,  as  brk., 
broken,  sml.,  small,  etc. .  A  combination  of  these  placed  by  a 
sounding  shows  at  once  the  material,  color,  and  nature. 

Buoys. — These  are  indicated  thus:  B.,  black;  E.,  red; 
H.  S.,  horizontal  stripes,  black  and  red,  danger  buoy; 
V.  S.,  vertical  stripes,  black  and  white,  channel  buoy.  1ST 
means  a  nun  buoy,  C  a  can  buoy,  S  a  spar  buoy.  On  enter- 
ing a  harbor,  black  buoys  are  left  on  the  port  hand,  red  on 
the  starboard  hand.  Black  buoys  have  odd,  red  buoys  even 
numbers.  Buoys  with  perch  and  square,  or  with  perch  and 
ball,  are  often  found  at  turning  points.  There  are  also  bell 
and  whistling  buoys,  lighted  (gas  or  electric)  buoys,  and 
white  anchorage  buoys.  Yellow  buoys  are  used  to  mark  quar- 
antine grounds  or  stations. 

Dangers. — Eock  awash  at  low  water,  *  ;  sunken  rock,  +. 
Dangers  of  doubtful  existence,  marked  E.  D. ;  if  known, 

but  of  doubtful  position,  marked  P.  D.     Anchorage,    HK   ; 

.a  wreck,    J^    or     1 1 1    ;   light  ship,      T  T 

Lights. — Light  houses  are  indicated  by  a  yellow  spot  with 
a  red  or  black  dot,  or  as  shown  in  Plates  X  and  XI,  end  of 
book.  Visibility  is  for  a  height  of  eye  of  15  feet  above  the 
sea  level. 

NOTE.— Symbols  on  charts  vary  according  to  the  origin  of  the  charts.  See  Plates 
X  and  XI. 


36  NAVIGATION 

Character  of  light. — Indicated  by  abbreviations: 

Lt.  F.  W.— A  fixed  steady  light,  white. 

Lt.  Fig.  E. — Short  flashes,  longer  intervals,  color  red. 

Lt.  Int.  R. — Long  flashes,  short  intervals,  color  red. 

Lt.  Eev.  W.- — Intensity  gradually  increasing  and  decreas- 
ing, color  white. 

Lt.  F.  and  Fig. — Combined  fixed  and  flashing. 

Currents. — These  are  indicated  by  feathered  arrows  point- 
ing in  general  direction  of  set,  with  figures  to  indicate  drift  in 
knots  per  hour;  current  flood,  by  an  unfeathered  arrow  with 
one,  two,  or  three  cross  marks  for  1st,  2d,  or  3d  quarter  of 
flow,  with  figures  to  indicate  velocity  in  knots  per  hour; 
current  ebb,  as  for  flood,  using  an  unfeathered  half  arrow; 

31.  Use  of  charts. — Spread  the  chart  out  before  you  on  the 
chart  board  with  the  North  direction  away  from  you ;  in  this 
way  no  readings  will  be  upside  down.  In  connection  with 
the  chart  a  navigator  requires  the  use  of  a  pair  of  parallel 
rulers,  a  pair  of  dividers,  a  sharp  pencil,  a  reading  glass,  and 
sometimes  a  course  protractor.  The  parallel  rulers  are  used 
to  transfer  a  course  or  bearing  from  the  compass  rose  so  as 
to  pass  it  through  a  given  point,  or  to  transfer  a  line  passing 
through  a  given  point  to  the  compass  rose  in  order  to  ascer- 
tain the  true  or  magnetic  bearing  or  course;  the  dividers  are 
used  for  taking  off  and  measuring  distances,  whilst  both  are 
used  in  plotting  or  taking  off  the  latitude  and  longitude  of  a 
point. 

To  find  the  latitude  of  a  place  on  a  Mercator  chart. — Bring 
the  edge  of  the  parallel  rulers  to  pass  through  the  place  par- 
allel to  a  parallel  of  latitude;  where  it  cuts  the  graduated 
meridian  on  the  chart's  side  is  the  latitude. 

To  find  the  longitude. — Bring  the  edge  of  ruler  to  pass 
through  the  place  parallel  to  a  meridian;  where  it  cuts  the 
graduated  parallel  at  top  or  bottom  of  chart  is  the  longitude. 


NAVIGATIONAL  INSTRUMENTS  37 

To  plot  a  given  latitude  and  longitude  on  a  Mercator 
chart. — Place  edge  of  ruler  along  the  parallel  of  latitude 
nearest  given  latitude,  move  ruler  parallel  to  itself  till  edge 
passes  through  given  latitude  on  the  graduated  meridian,  hold 
it  firmly  to  prevent  slipping;  with  dividers  take  from  upper 
or  lower  graduated  margin  the  distance  of  given  longitude 
from  nearest  meridian,  and  lay  it  off  from  the  same  meridian 
along  the  edge  of  the  parallel  rulers.  Or,  in  the  absence  of 
dividers,  with  a  pencil  point  draw  a  light  line  along  edge  of 
ruler  across  approximate  longitude;  then  lay  the  ruler  par- 
allel to  the  meridian,  the  edge  cutting  the  longitude  scale 
at  the  proper  longitude,  and  cros^  the  above  line  along  the 
ruler's  edge;  the  intersection  is  the  plotted  position. 

On  a  polyconic  chart,  positions  are  plotted,  or  taken  off, 
less  accurately;  the  graduated  parallel  and  meridian  of  that 
graduated  subdivision  nearest  the  position  being  used. 

To  measure  a  distance  between  two  points  on  a  Mercator 
chart. — In  whatever  way  the  distance  may  run,  take  off  the 
distance  with  a  pair  of  dividers  and  measure  it  along  the  grad- 
uated meridian  or  latitude  scale,  so  that  the  middle  of  the 
line  will  be  in  the  middle  latitude  between  the  two  points; 
for  instance,  on  chart,  Fig.  12,  the  line  AB  should  be  measured 
so  that  its  middle  point  g  will  be  over  h.  In  case  the  distance 
runs  E.  and  W.  on  a  parallel,  then  the  distance  should  be 
measured  equally  each  side  of  the  parallel;  for  instance,  on 
the  same  chart  as  above  (Fig.  12),  the  distance  BC  should  be 
so  applied  to  the  latitude  scale  that  its  middle  point  would  be 
over  Tc.  In  case  the  distance  is  too  great  to  be  conveniently 
included  between  the  points  of  the  dividers,  take  with  the 
dividers  a  convenient  unit  from  the  latitude  scale  so  that  the 
middle  latitude  will  be  about  midway  between  the  points  of 
the  dividers,  then  step  off  this  unit  along  the  distance  to  be 
measured,  turning  the  dividers  alternately  to  right  and  to 
left,  counting  the  number  of  times  the  unit  is  contained  in 


38  NAVIGATION 

the  distance.  The  unit,  which  may  be  5,  10,  or  any  number 
of  miles,  multiplied  by  the  number  of  times  it  is  stepped  off, 
plus  any  fraction  of  the  unit  (measured  in  its  own  middle 
latitude)  to  the  end  of  the  line,  will  give  the  required  distance. 
In  making  the  above  measurements  the  middle  parallel  is 
never  drawn  but  is  assumed  by  inspection. 

In  measuring  distances  on  a  polyconic  chart,  reference  is 
made  not  to  the  margins  of  the  chart  but  to  the  single  scale 
of  distance  under  the  title  of  the  chart. 

To  find  the  course  from  one  point  to  another  on  the  Mer- 
cator  chart. — Lay  down  the  ruler  so  its  edge  passes  through 
the  two  points,  and  draw  a  line  if  desired.  Now  move  the 
ruler  parallel  to  itself  till  the  same  edge  passes  through  the 
nearest  compass  rose  and  read  the  course  or  bearing  from  the 
diagram.  Or,  having  drawn  the  line  in  the  first  place,  meas- 
ure with  a  protractor  the  angle  it  makes  with  any  meridian. 
When  the  diagram  is  constructed  with  reference  to  the  true 
meridian,  its  readings  indicate  the  true  course,  otherwise  the 
magnetic  course. 

Exactly  the  same  method  of  procedure  is  followed  in  find- 
ing the  course  on  a  polyconic  chart,  but,  from  the  nature  of 
the  projection,  it  is  evident  that  this  straight  line  is  not  a 
rhumb  line,  and  that  the  course  must  be  changed  after  a  time, 
on  account  of  the  angle  between  the  meridians ;  the  length  of 
the  time  depending  on  the  general  bearing  between  the  points, 
on  the  distance,  on  the  latitude,  and  on  the  scale  of  the  chart. 

The  course  and  distance  run  by  a  ship  on  the  rhumb  line 
from  a  given  point  being  known,  to  find  the  ship's  position 
on  the  mercator  or  polyconic  chart. — Place  the  edge  of  the 
parallel  rulers  so  as  to  pass  through  the  center  of  the  com- 
pass rose  and  the  reading  of  its  circumference  representing 
the  course  (true  or  magnetic).  Move  the  ruler  parallel  to 
itself  till  the  same  edge  passes  through  the  given  point. 
Draw  a  light  line  in  the  desired  direction  and  lay  off  the 


NAVIGATIONAL  INSTRUMENTS  39 

distance  run  from  the  given  point  on  this  line,  or,  along  the 
edge  of  the  ruler,  if  the  line  is  not  drawn,  and  the  ship's  place 
is  determined.  The  distance  is  taken  from  the  proper  scale 
as  explained  in  previous  paragraphs  for  both  the  mercator 
and  polyconic  charts. 

To  plot  the  ship's  position  by  cross  bearings. — Correct  each 
bearing  for  the  deviation  of  the  compass  due  to  the  direction 
of  the  ship's  head  when  bearing  was  taken ;  the  magnetic  bear- 
ings are  thus  obtained.  Place  the  edge  of  the  parallel  rulers 
over  a  magnetic  compass  rose,  the  edge  passing  through  the 
center  and  reading  of  the  circumference  representing  the 
magnetic  bearing.  Move  the  ruler  parallel  to  itself  till  the 
same  edge  passes  through  the  proper  object,  draw  a  light  line 
through  the  approximate  position  of  the  ship.  This  line  is  a 
line  of  bearing  and  the  position  of  the  ship  is  somewhere  on 
it.  In  the  same  way  draw  the  line  of  bearing  corresponding 
to  the  second  object.  The  ship  being  on  both  lines  will  be  at 
their  intersection  on  the  chart.  To  obtain  good  cuts,  these 
lines  should  make  angles  not  less  than  30°,  the  best  cuts,  of 
course,  being  given  when  the  lines  are  at  right  angles  to  each 
other.  If  the  compass  rose  is  a  true  and  not  a  magnetic  rose, 
the  bearings  must  be  corrected  for  the  variation  as  well  as  the 
deviation. 

32.  Correction  of  charts.1 — Charts,  to  be  of  any  service, 
should  be  reliable,  and  to  be  reliable  they  must  be  kept  cor- 
rected to  date.     The  information  for  this  purpose  can  be  got- 
ten from  "  Notices  to  Mariners/'  bulletins  published  weekly 
by  the  Hydrographic  Office  of  the  Navy  Department,  and 
the  Light-House  Board;  also  from  the  branch  hydrographic 
offices  at  our  important  sea  ports. 

33.  Arrangement  and  stowage  of  charts. — The  U.  S.  Hy- 
drographic Office  issues  to  ships  of  the  navy  Hydrographic 
Office  (H.  0.),  Coast  Survey  (C.  S.),  and  British  Admiralty 
(B.  A.)  charts.     Eegardless  of  the  publication  or  chart  num- 

1  The  U.  S.  Naval  Wireless  Telegraph  Stations  on  the  seaboard  transmit  daily  at 
6  a.  m.,  2  p.  m.,  and  10  p.  m.,  standard  time,  from  the  Hydrographic  Office  to 
vessels  at  sea,  information  as  to  obstructions  that  are  dangerous  to  navigation. 


40  NAVIGATION 

ber,  all  charts  issued  are  arranged  as  far  as  practicable  in  geo- 
graphical sequence,  numbered  consecutively,  and  divided  into 
portfolios,  each  portfolio  containing  about  100  charts.  The 
consecutive  numbers  in  each  portfolio  begin  with  the  even 
hundred;  a  chart  whose  consecutive  number  is,  for  instance, 
520,  will  be  found  in  portfolio  No.  5. 

General  charts  of  the  ocean  will  be  found  in  portfolio  No.  1. 
General  charts  of  the  station  for  the  use  of  the  commander-in- 
chief  will  be  found  in  portfolio  No.  10;  those  for  the  use  of 
the  wardroom  officers  in  portfolio  No.  11.  Each  portfolio 
should  have  a  separate  drawer,  in  a  nest  of  drawers,  built  in 
the  pilot  house  and  convenient  to  the  chart  table. 

34.  The  three-arm  protractor. — In  determining  the  position 
of  a  ship  by  sextant  angles  between  known  objects  along  a 
coast,  the  three-arm  protractor  will  prove  itself  an  invaluable 
instrument.  It  consists  of  a  graduated  brass  circle  having 
three  arms,  the  straight  edges  of  which  all  pass  through  the 
center  of  the  circle.  The  center  arm  is  fixed  and  the  zero  of 
graduation  is  coincident  with  its  straight  edge.  The  other 
two  arms  are  movable  and  both  are  fitted  with  clamp  screws 
and  tangent  screws.  As  the  movable  arms  turn  away  from 
the  central  arm,  the  angles  gradually  increase,  and  when  the 
arms  are  clamped,  a  vernier,  with  reading  microscope,  gives 
the  angle  to  the  least  count  of  the  vernier.  Extension  pieces 
are  provided  for  each  arm.  It  is  impossible  to  shut  the  right 
arm  close  home,  as  the  beveled  straight  edge  of  the  fixed 
arm  is  on  its  left  side,  so  if  the"  right  arm  can  not  be  set  for 
a  small  right  observed  angle,  set  the  left  arm  for  it;  then 
swing  the  right  arm  around  and  set  it  for  the  sum  of  the  two 
observed  angles,  reading  from  zero  to  the  left. 

To  plot  a  vessel's  position  with  a  three-arm  protractor. — 
Select  three  objects  that  can  be  seen  and  reflected,  that  are 
well  located  on  the  chart,  and  so  situated  with  reference  to 
each  other  that  the  observer's  position  will  be  well  determined. 


NAVIGATIONAL  INSTRUMENTS  41 

Get  simultaneously  the  angle  between  the  middle  object  and 
the  right  one  (called  the  right  angle),  and  the  angle  between 
the  middle  object  and  the  left  one  (called  the  left  angle). 
The  lateral  arms  of  the  protractor  having  been  set  to  their 
proper  angles,  and  the  same  verified,  the  instrument  is  placed 
on  the  chart,  the  edge  of  the  central  arm  passing  through  the 


The  Three-Arm  Protractor. 

middle  object  and  kept  there  whilst  the  instrument  is  moved 
around  till  the  edges  of  the  lateral  arms  also  pass  through 
their  respective  objects.  The  center  of  the  instrument  is  at 
the  point  of  observation  which  is  lightly  marked  upon  the 
chart  by  pencil  or  the  spring  point  of  the  center  punch. 

Tracing  paper  or  linen  with  angles  laid  off  and  properly 
numbered  may  be  used  as  a  substitute. 

The  diagram  (Fig.  13)  will  illustrate  the  different  cases 


42  NAVIGATION 

that  may  be  met  with  in  practice.  A,  By  C  are  the  three  ob- 
jects forming  the  triangle  called  the  great  triangle,  the  circle 
through  which  is  called  the  great  circle.  The  position  of  the 
observer  is  at  the  intersection  of  the  circles  of  which  the  sides 
of  the  great  triangle  are  chords,  the  position  of  the  centers 


of  these  circles,  and  hence  of  their  intersection,  depends  on 
the  observed  angles.  The  nearer  these  secondary  circles  inter- 
sect at  90°,  the  better  the  "  fix/'  In  cases  in  which  the  cen- 
ters of  the  circles  are  near  each  other,  and  near  the  center  of 
the  great  circle,  the  position  is  more  or  less  indeterminate, 
and  such  angles  are  called  "  revolvers." 


NAVIGATIONAL  INSTRUMENTS  43 

CASE  1. — The  two  angles  observed  are  >180°;  the  position 
of  observer  is  within  the  triangle  and  is  well  determined. 

CASE  2. — The  sum  of  the  two  angles  =  180° ;  the  observer 
is  on  one  side  of  the  great  triangle,  and  the  position  is  well 
determined. 

CASE  3. — A  range  and  one  angle;  a  good  determination  of 
position. 

CASE  4. — The  middle  object  is  nearer  than  the  other  two; 
the  position  can  be  determined  very  well,  but  A  should  not 
be  so  close  as  to  make  angles  too  small,  small  angles  making 
position  uncertain. 

CASE  5. — Using  three  objects  in  line  or  nearly  so,  as  in 
the  case  of  objects  5,  A',  and  0.  An  excellent  arrangement; 
the  larger  the  angles,  the  more  reliable  the  "  fix/' 

CASE  6.— Where  the  sum  of  the  observed  angles  is  the 
supplement  of  BAG;  the  position  is  indeterminate  as  it  may 
be  anywhere  on  the  great  circle. 


CHAPTEE  III. 
NAVIGATIONAL  INSTRUMENTS. 

The  Compass  and  Pelorus. — Compass  Error.— 
Theory  of  Deviations. 

SECTION  I. 

35.  The  mariner's  compass  is  one  of  his  most  important 
and  essential  instruments,  showing  him  how  he  is  steering, 
enabling  him  to  direct  his  ship  on  a  desired  course,  or  to  get 
bearings  of  objects  in  sight  from  which  to  determine  his  po- 
sition. 

It  consists  essentially  of  a  needle,  or  a  series  of  needles,  of 
strong  and  powerful  magnetism,  attached  to  a  properly 
graduated  card  which  is  mounted  at  its  center  on  a  pivot  in 
the  center  of  the  compass  bowl,  and  has  free  movement  in  the 
horizontal  plane.  The  bowl  is  made  of  copper,  hemispheri- 
cal in  shape,  is  heavy  as  well  as  ballasted,  and  swings  on  knife 
edges  in  gimbals,  thus  enabling  the  card  to  maintain  a  hori- 
zontal position  even  in  a  seaway. 

Inside  the  bowl  are  painted  two  vertical  black  lines  180° 
apart,  the  one  towards  the  head  of  ship  being  called  the  lub- 
ber's line.  The  bowl  is  so  mounted  that  a  line  through  the 
pivot  and  the  lubber's  line  is  parallel  to  the  keel  line  of  the 
ship,  so  that  this  lubber's  line  indicates  the  course,  or  the 
direction  of  the  ship's  head  per  compass. 

'The  compass  card  is  divided  into  360°;  the  graduation  be- 
ginning with  0°  at  North  runs  around  to  the  right  and  is 
numbered  at  every  fifth  degree.  The  card  is  also  divided  into 
points  and  quarter  points  (see  Appendix  E). 


NAVIGATIONAL  INSTRUMENTS  45 

The  two  general  classes. — There  are  two  gerieral  classes  of 
compasses  in  use,  the  dry  and  the  liquid.  In  the  latter,  the 
bowl  is  filled  with  liquid  which,  together  with  the  hollow  card, 
gives  a  certain  amount  of  buoyancy  to  the  card  and  hence 
regulates  its  pressure  on  the  pivot  and  ease  of  movement,  and 
also,  through  its  inertia,  tends  to  prevent  or  reduce  vibrations 
due  to  the  ship's  motion. 

The  liquid  compass  is  used  in  the  U.  S.  Navy,  and,  accord- 
ing to  the  purposes  it  serves,  a  compass  is  designated  as 
service,  conning  tower,  or  boat  compass.  Though  made  on  the 
same  general  principles,  they  embody  different  degrees  of 
excellence  and  have  cards  of  different  sizes.  The  service  com- 
pass has  a  card  7-J  inches,  conning  tower  5  inches,  boat  com- 
pass 4  inches  in  diameter.* 

The  service  comDass  is  further  designated,  according  to  its 
use  and  location  on  board,  as  standard,  steering,  manoeuvring, 
battle,  auxiliary  battle,  top  and  check  compass. 

Location  of  standard. — The  standard  is  the  compass  by 
which  the  ship  should  be  navigated,  all  others  being  regarded 
as  auxiliaries,  as  for  the  use  of  helmsmen,  etc. 

It  should  be  placed  in  the  midship  line  of  the  ship,  at  a 
position  where  the  mean  directive  force  is  a  maximum,  if  pos- 
sible; as  far  removed  as  practicable  from  considerable  masses 
of  iron,  especially  if  vertical,  the  influences  of  the  dynamos 
or  electrical  currents,  stands  of  arms,  or  other  iron  or  steel 
subject  to  occasional  removal.  It  should  be  mounted  at  least 
five  feet  from  an  iron  deck  or  beams,  in  a  compensating  bin- 
nacle, easily  accessible  at  all  times,  conveniently  near  the 
steering  compass,  and  so  located  that  all  around  bearings  of 
land  or  heavenly  bodies  can  be  observed. 

36.  The  service  or  7^"  liquid  compass. — This  compass  con- 
sists of  a  tinned  brass  skeleton  card  7-J  inches  in  diameter. 
It  is  of  a  curved  annular  type,  the  outer  ring  convex  on  the 
upper  and  inner  side,  graduated  to  read  to  quarter  points, 

*  The  compasses  in  submarines  are  of  special  types,  usually  furnished  by  the  con- 
tractors to  suit  the  special  conditions.  As  a  rule  they  are  transparent  and  set  in  the 
deck  so  as  to  be  read  either  from  inside  or  outside  of  the  boat,  reflecting  prisms  and 
lenses  being  used  where  necessary. 


46  NAVIGATION 

with  the  outer  edge  divided  to  half  degrees,  and  figured  at 
each  fifth  degree  from  0°  at  North,  numbering  to  the  right 
through  360°.  The  card  has  a  concentric  spheroidal  air 
vessel,  to  assist  in  giving  buoyancy  to  the  card  and  magnets,  so 
that  the  pressure  on  the  pivot  at  60°  F.  will  vary  between 
60  and  90  grains.  The  air  vessel  has  a  hollow  cone,  open  at 
the  lower  end,  carrying  a  sapphire  cap  at  the  apex,  by  which 
the  card  is  supported  on  the  pivot. 

The  magnets,  four  in  number,  consist  of  cylindrical  bundles 
of  steel  wires,  each  .06  of  an  inch  in  diameter,  strongly  mag- 
netized, put  into  a  sealed  cylindrical  case  and  secured  to  the 
card  parallel  to  its  North  and  South  diameter.  The  cases  of 
two  of  the  magnets,  each  magnet  5J"  long,  pass  through  the 
air  vessel  to  which  they  are  soldered,  and  have  their  ends  se- 
cured to  the  bottom  of  the  card  ring,  like  ends  on  chords  of 
nearly  15°  passing  through  their  extremities.  The  other  two 
cases  containing  magnets,  each  4f  inches  long,  are  placed  par- 
allel to  the  longer  magnets,  on  chords  of  nearly  45°  of  a  circle 
through  the  extremities,  and  the  ends  are  secured  to  the 
bottom  of  the  card  ring. 

The  card  is  mounted  in  a  bowl,  made  of  cast  bronze,  on  a 
bell-metal  pivot  fastened  to  the  center  of  the  bottom  of  the 
bowl  by  a  flanged  plate  and  screws.  Through  this  plate  and  the 
bottom  of  the  bowl  are  two  small  holes  which  communicate 
with  a  metallic  self-adjusting  expansion  chamber  located  just 
beneath  the  bowl.  These  holes  permit  a  circulation  of  liquid 
between  the  bowl  and  expansion  chamber,  and  it  is  the  func- 
tion of  the  latter  to  keep  the  bowl  full  of  liquid  without  show 
of  bubbles,  or  undue  pressure  that  might  be  caused  by  change 
in  the  volume  of  the  liquid  due  to  changes  of  temperature. 

The  liquid  used  is  composed  of  45  per  cent  pure  alcohol  and 
55  per  cent  distilled  water,  and  remains  liquid  at  a  tempera- 
ture lower  than  — 10°  F.  The  inside  of  the  bowl  is  painted 
white  with  a  paint  insoluble  in  the  above  liquid.  An  enam- 
eled plate  is  secured  on  the  inside  of  the  bowl,  and  on  this 
plate  a  lubber's  line  is  drawn. 

The  bowl  is  fitted  with  a  glass  cover,  the  edge  of  which  is 


NAVIGATIONAL  INSTRUMENTS  47 

closely  packed  with  rubber,  completely  preventing  leakage  or 
evaporation  of  the  liquid,  which  at  all  times  fills  the  bowl. 
The  rim  of  the  compass  bowl  is  made  rigid  and  its  outer  and 
upper  edges  turned  accurately,  that  the  service  azimuth  circle 
when  in  use  may  be  properly  seated.  The  bowl  has  a  false 
bottom  containing  a  leaden  weight  as  ballast  to  keep  the  bowl 
horizontal. 


FIG.  14. — United  States  Navy  Standard  Compass. 

As  made  by  E.  S.  Ritchie  &  Sons. 

The  compass  is  mounted  in  gimbals  with  knife-edge  bear- 
ings in  its  binnacle.* 

37.  The  IT.  S.  Navy  standard  compensating  binnacle. — 
The  binnacle  stand  includes,  in  a  single  brass  casting,  the 
circular  base,  cylindrical  pedestal,  conical  magnet  chamber, 
cylindrical  compass  chamber,  and  the  graduated  arms  for  the 
quadrantal  correctors  at  right  angles  to  the  keel  line  of  the 
binnacle  (see  Fig.  15). 

*  A  new  compass,  having  both  card  and  bottom  of  bowl  transparent,  with  electric 
illumination  from  below,  is  to  be  issued.  The  5%"  card  is  much  smaller  than  bowl. 
The  lubber's  point  is  a  pointer  from  side  of  bowl  to  card.  Besides  expansion  chamber 
and  weight,  the  bowl  has  an  annular  ring  containing  oil  as  an  oscillation  damper. 


48  NAVIGATION 

The  hood  is  spun  of  stout  polished  brass,  has  a  hinged  plate- 
glass  front  opening  upwards,  and  a  sliding  door  opposite  the 
glass  to  permit  bearings  to  be  taken  in  wet  weather  without 
removal  of  the  hood.  Clips  on  the  binnacle  take  over  the 
rounded  edge  of  the  binnacle  hood  and  hold  it  on.  The  hood 
carries  in  its  center  a  lamp  provided  with  a  prism  for  reflect- 
ing on  the  compass  dial  the  light  of  a  5  c.  p.  electric  light. 
The  plug  for  this  light  is  in  the  pedestal  below. 

The  rectangular  method  *  of  compensation  is  used  in  cor- 
recting the  semicircular  deviation.  The  correcting  magnets  are 
mounted  on  trays  which  can  be  raised  or  lowered,  indepen- 
dently of  each  other,  by  a  screw  moved  by  beveled  gears,  and 
so  constructed  that  they  will  pass  each  other  in  any  position, 
the  mechanism  permitting  an  extent  of  travel  of  12  inches. 

The  semicircular  magnets  are  held  in  their  receptacles  by 
a  spring-closing  device,  each  carrier  or  tray  having  a  group  of 
three  magnets  each  side  of  the  vertical  axis,  making  six  in  all, 
the  tubes  being  horizontal,  one  over  the  other. 

The  quadrantal  correctors  are  removable  soft  iron  spheres, 
secured  to  the  brackets  by  screw  bolts,  the  centers  of  the 
spheres  lying  in  the  same  horizontal  plane  as  the  compass 
needles.  They  are  capable  of  motion  towards  or  from  the 
compass,  the  distance  from  whose  center  is  indicated  by  a 
scale  in  inches  and  quarter  inches  on  each  arm. 

The  heeling  corrector  consists  of  a  cylindrical  magnet 
having  a  hook  in  each  end  to  which  is  attached  a  chain.  Cen- 
trally, in  the  vertical  axis  of  the  binnacle,  there  is  a  hollow 
brass  tube  extending  the  entire  depth  of  stand  from  the  bot- 
tom of  compass  chamber.  By  removing  the  compass,  the 
heeling  corrector  attached  to  its  chain  may  be  lowered  into 
this  tube,  and  held  at  the  proper  height  by  the  chain  which 
passes  over  a  roller  at  the  top  of  the  tube  and  secures  to  a 
cleat,  or  by  a  set  screw  in  the  magnet  chamber. 

*  By  fore  and-aft  and  athwartship  magnets  (Art.  81,  82.) 


NAVIGATIONAL  INSTRUMENTS  49 


38.  The  gyro-compass. — This  consists  essentially  of  a  gyroscope, 
its  axis  being  horizontal  when  started,  mounted  below  an  annular 
or  ringed-shaped  bowl  of  mercury,  in  which  the  float  carrying  the 
gyroscope,  with  compass  card  attached  above,  is  suspended.  The 
bowl  is  supported  by  gimbal  rings  so  hung  within  the  binnacle  as 
to  prevent  vibration.  The  gyroscope  being  spun  to  about  20,000 
revolutions  per  minute  by  a  motor,  the  rotor  of  which  is  the  gyro, 
gives  a  very  strong  directive  force. 

Briefly  stated,  this  arrangement  is  first  a  pendulum  tending  to 
hang  vertically,  and  second,  when  the  gyro  is  rotating,  it  is  a 
gyroscope  tending  to  maintain  its  fixed  plane  of  rotation.  It  is 
the  combination  of  these  two  efforts  that  causes  it  to  point  to  the 
North. 

This  may  be  readily  understood  by  considering  its  action  as 
follows : 

The  gyro  being  set  in  motion  with  the  pendulum  hanging 
vertically,  there  is  at  first  no  force  acting  to  turn  it  from  its 
original  plane  of  rotation.  But  should  the  axis  of  rotation  be  in 
any  other  direction  than  North  and  South,  suppose  East  and  West, 
the  rotation  of  the,  earth  would  in  a  few  minutes  put  the  pendulum 
out  of  plumb,  provided  the  gyro  were  able  to  retain  its  original 
plane  of  rotation.  The  pull  of  gravity  endeavoring  to  bring  the 
pendulum  vertical,  and  the  pull  of  the  gyro  endeavoring  to  retain 
its  original  plane,  cause  a  new  movement  about  a  vertical  axis, 
which  turns  the  axis  of  rotation  of  the  gyro  into  the  meridian. 

Being  independent  of  the  magnetism  of  both  earth  and  ship, 
the  gyro-compass  is  well  adapted  to  use  in  locations  behind  armor, 
in  close  proximity  to  magnetic  material  and  stray  magnetic  fields, 
and  hence  its  special  use  is  as  a  battle  compass. 

By  means  of  proper  transmitting  mechanism,  the  indications  of 
a  "  Master  Compass "  located,  say  in  the  central  station,  or 
wherever  most  convenient,  without  regard  to  surrounding  mag- 
netic conditions,  may  be  shown  by  "  Repeater  Compasses  "  elec- 
trically connected  with  it  and  installed  in  the  conning  tower, 
plotting  room,  steering  engine  room  and  fire-control  tower. 

The  gyro-compass  must  be  compensated  in  any  given  latitude  for 
deviations  caused  by  speed  and  course,  also  for  errors  due  to 
marked  changes  of  latitude.  Tables  giving  the  correction  in  each 
case  are  supplied  with  the  instrument  which  is  fitted  with  a 
movable  lubber's  point  for  compensating  purposes. 

As  the  frictional  resistance  to  rotation  about  a  vertical  axis  is 
made  a  minimum,  the  axis  of  the  gyro  does  not  of  itself  steady  on 
North,  but  oscillates  each  side  of  the  meridian.  A  special  device 
is  used  to  damp  the  oscillations,  which  is  not  effectually  done  in 
less  than  one  hour  and  a  half  after  first  starting. 

This  fact  and  the  further  fact,  that  on  a  marked  change  of 
course,  the  gyro  does  not  settle  down  for  several  minutes  on  the 
new  course,  are  the  greatest  drawbacks  to  its  use  as  a  battle 
compass. 

However,  for  all  purposes  of  navigation,  when  it  is  not  necessary 
for  the  personnel  to  remain  behind  armor,  the  magnetic  compass, 
mounted  in  the  open  and  properly  compensated,  is  superior. 


FIG.  15. — U.  S.  Navy  Standard  Compensating  Binnacle. 
As  made  by  Keuffel  &  Esser  Co. 


NAVIGATIONAL  INSTRUMENTS  51 

39.  Azimuth  circle. — This  consists  of  a  composition  ring 
turned  true  to  fit  the  compass  bowl.  At  one  extremity  of  a 
diameter  is  a  curved  mirror  hinged  to  move  around  a  horizon- 
tal axis  and  facing  at  the  other  extremity  a  prism  completely 
encased  in  a  brass  case  except  for  a  narrow  vertical  slit.  The 
sun's  rays  reflected  by  the  mirror  upon  the  slit  appear  as  a 
thin  pencil  of  light  on  the  graduations  of  the  card  circle.  A 
level  on  the  rim  serves  to  show  when  it  is  horizontal. 


FIG.  16.— U.  8.  Navy  Azimuth  Circle. 

(To  fit  Navy  Standard  Compass  No.  1.) 

As  made  by  E.  S.  Ritchie  &  Sons. 

A  second  set  of  vanes,  at  the  extremities  of  a  diameter  at 
right  angles  to  the  first,  is  used  for  direct  bearings  of  distant 
objects,  of  stars,-  or  of  the  sun  when  partially  obscured.  At 
one  end  is  a  mirror  reflecting  the  image  of  the  sun  or  star, 
and  a  prism  reflecting  the  card  circle  and  vane  simultaneously 
to  the  eye  at  the  other  end  (Fig.  16). 

40.  The  pelorus  (Fig.  17). — This  is  an  instrument  located 
on  board  ship  at  some  point  where  a  clear  view  can  be  ob- 
tained for  taking  bearings.  It  is  most  convenient  to  have 
one  at  each  end  of  the  flying  bridge.  It  consists  of  a  circu- 


NAVIGATION 


lar  plate  mounted  in  gimbals  whose  knife  edges  rest  in  the 
Y's  of  a  vertical  standard  rigidly  secured  in  place. 

The  circular  plate  has  a  raised  flange  on  its  periphery  with 
heavy  marks  90°  apart  to  correspond  with  the  fore-and-aft 
and  athwartship  lines  of  the  ship. 

Concentric  with  the  plate 
and  each  other  are  a  dial  plate 
and  an  alidade,  each  capable  of 
independent  movement  in  azi- 
muth. The  dial  plate  is 
simply  a  dumb  compass  card 
of  metal,  graduated  to  quarter 
points  and  single  degrees, 
whose  upper  surface  is  flush 
with  the  raised  periphery,  and 
is  provided  with  a  clamp.  The 
alidade  is  fitted  with  a  level, 
folding  sight  vanes,  hinged  re- 
flector, and  a  sliding  peep  sight 
with  neutral  glass.  The  line 
of  sight  through  the  vanes 
passes  through  the  vertical  axis 
of  the  instrument  and  is  indi- 
cated on  the  dial  plate  by  an  in- 
dex at  each  end  of  the  alidade. 
The  alidade  is  also  fitted  with 
a  clamp.  A  heavy  balance 
weight  is  attached  to  the  lower 
center  of  the  plate.  It  may  be 
used  to  eliminate  the  compass  error  from  observed  bearings 
by  setting  the  alidade  to  a  reading  which  is  the  compass  course 
corrected  for  the  error,  and,  as  long  as  the  ship  is  on  that 
particular  heading  for  which  the  dial  is  set,  all  bearings  by 
the  pelorus  will  be  true;  if  correction  is  made  for  deviation 
only,  then  the  bearings  will  be  magnetic. 


FIG.  17. 


PLATE  I. 


THE  ILLUMINATED  DIAL  PELOKUS. 


PLATE  II. 


TUB  ILLUMINATED  DIAL  PELORUS- 
CHAMBER  AND  BOWL. 


54  NAVIGATION 

41.  Illuminated   dial   pelorus    (Plates   I   and   II).— The 
pelorus  standard  consists  of  three  parts,  the  base,  supporting 
column,  and  pelorus  chamber;  on  the  top  edge  of  the  latter 
are   scored   the   fore-and-aft   and   athwartship    marks.     The 
pelorus  bowl  is  mounted  on  two  trunnions  in  a  gimbal  ring 
which  pivots  on  the  athwartship  diameter  of  pelorus  chamber. 
The  pelorus  bowl  is  built  up  of  a  top  bowl  ring  and  a  shell 
of  sheet  brass.     A  seat  is  turned  in  the  top  bowl  ring  to  re- 
ceive the  assembled  pelorus  card,  the  principal  part  of  which 
consists  of  a  disk  of  clear  plate  glass  9  inches  in  diameter  by 
3^-of  an  inch  thick.     The  card  is  graduated  in  degrees  near 
its  outer  edge,  every  fifth  degree  accentuated,  and  every  tenth 
degree  is  marked  in  figures.     Graduations  run  from  0°  at  N". 
to  360°  around  to  right,  and,  inside  these  degree  marks,  the 
card   is   graduated   to    quarter    points.     An    azimuth    circle 
marked  in  degrees  is  permanently  secured  to  the  bowl  ring. 
The   pelorus   card   has   unobstructed   rotation   except   when 
clamped  by  a  clamping  screw,  which  is  at  the  90°  graduation 
of  the  azimuth  circle.     There  is  an  alidade  capable  of  free 
revolution  either  way,  or  of  being  securely  clamped ;  it  is  pro- 
vided with  a  level,  folding  sights,  hinged  reflector,  and  peep 
sight.     This  pelorus  is  designed  to  be  used  at  night  without  a 
navigator's  lantern,  the  transparent  dial  being  illuminated  by 
an  electric  light  placed  beneath  it  in  the  standard. 

42.  The  use  of  pelorus  to  determine  a  magnetic  heading. — 
To  place  the  ship's  head  on  any  magnetic  point  by  the  pelorus : 

(1)  With  the  known  latitude  of  place  and  declination  of 
body — say  the  sun— find  from  the  azimuth  tables  the  sun's 
true  bearing  for  certain  selected  local  apparent  times.     From 
these  true  bearings  find  the  sun's  magnetic  bearings  for  the 
same  times  by  applying  the  variation  of  the  locality,  easterly 
variation  being  applied  to  the  left,  westerly  variation  to  the 
right  of  the  true  azimuth. 

(2)  Shortly  before  the  earliest  time  selected,  set  that  point 


COMPASS  CORRECTIONS  55 

of  the  pelorus  corresponding  to  the  magnetic  heading  desired 
on  the  forward  keel  line  or  indicator  and  clamp  the  plate ; 
set  the  sight  vanes  to  correspond  with  the  sun's  magnetic 
bearing  at  the  selected  time  and  clamp  the  vanes  to  the  plate. 
(3)  By  the  use  of  engines  and  helm,  bring  the  sight  vanes 
on  the  sun  and  keep  them  there,  being  careful  not  to  disturb 
the  clamps  of  plate  or  vanes,  noting  at  the  instant  of  the 
selected  local  apparent  time  the  heading  per  compass.  That 
heading  per  compass  corresponds  to  the  magnetic  direction 
desired. 

SECTION  II. 

43.  Compass    corrections. — The    compass    needle    seldom 
points  to  the  true  North,  so  in  order  to  obtain  a  true  course 
or  a  true  bearing,  certain  corrections  must  be  applied  to  the 
compass  course  or  the  compass  bearing,  as  the  case  may  be. 
They  consist,  according  to  circumstances,  of  one  or  more  of  the 
following;  i.  e.,  variation,  deviation,  leeway.     Each  of  these 
terms  will  be  explained  at  the  proper  time. 

44.  The  earth's  magnetism. — The  earth  is  a  huge,  natural 
but  irregular  magnet,  having  a  resultant  pole  in  each  hemi- 
sphere which,  however,  are  not  coincident  with  the  geograph- 
ical poles,  and  a  magnetic  equator  which  does  not  coincide 
with  the  geographical  equator.     The  North  magnetic  pole  is 
approximately  in  Lat.  70°  N,  Long.  96|°  W,  the  South  mag- 
netic pole  in  Lat.  73J°  S,  Long.  147J°  E.     Eecognizing  the 
laws  of  attraction  and  repulsion  between  two  bar  magnets, 
and  the  analogy  that  exists  between  the  magnetic  character 
of  the  earth  and  a  bar  magnet,  it  is  evident  that  the  mag- 
netism of  the  North  magnetic  pole  is  of  an  opposite  kind  to 
that  of  the  North-seeking  end  of  a  magnetized  needle;  there- 
fore, if  we  regard  the  magnetism  of  the  North  seeking  end 
of  the  needle  as  North  magnetism,  we  must  consider  the  North 
magnetic  pole  as  having  South  polarity  and  the  South  mag- 
netic pole  as  having  North  polarity. 


56  NAVIGATION 

However,  physicists  do  not  agree  as  to  which  shall  be  called 
North  magnetism,  that  of  the  North-seeking  end  of  the  needle 
or  that  of  the  North  magnetic  pole,  so  it  is  convenient  to  dis- 
tinguish them  by  colors,  calling  the  first  red  and  the  second 
blue. 

The  general  effect  of  the  earth's  magnetism  is  to  draw  the 
North  end  of  the  needle  towards  the  North  and  the  South 
end  towards  the  South ;  but,  with  the  exceptions  noted  further 
on,  a  freely  .suspended  magnetized  needle,  affected  only  by 
terrestrial  influences,  generally  speaking,  neither  points  in 
the  direction  of  the  true  meridian,  nor  lies  in  a  horizontal 
plane,  nor  occupies  the  same  relative  position  in  two  different 
places. 

Line  of  force  and  dip. — The  direction  in  which  the  needle 
does  point  at  any  given  place  is  the  "  line  of  total  magnetic 
force  "  at  that  place,  the  inclination  of  which  below  the  hori- 
zontal plane  is  called  the  "  magnetic  dip."  The  line  of  total 
force  is  spoken  of  as  the  "  line  of  force." 

The  magnetic  poles. — Those  two  positions  at  which  the 
line  of  total  force  is  vertical  are  known  as  the  magnetic  poles ; 
a  freely  suspended  magnetized  needle  would  be  vertical  at  the 
poles,  with  the  North  end  down  at  the  North  magnetic  pole 
and  the  South  end  down  at  the  South  magnetic  pole. 

The  magnetic  equator. — The  line  joining  all  those  posi- 
tions on  the  earth's  surface  at  which  the  "  line  of  force  "  is 
horizontal  is  known  as  the  magnetic  equator,  which  is  to  the 
northward  of  the  geographical  equator  in  the  Indian  Ocean 
and  the  western  half  of  the  Pacific  Ocean,  and  to  the  south- 
ward of  it  in  the  Atlantic  and  eastern  half  of  the  Pacific 
Ocean. 

As  we  have  magnetic  poles  and  a  magnetic  equator,  analo- 
gously we  have  magnetic  latitude;  all  points  of  the  earth  in 
North  magnetic  latitude  have  South  or  blue  magnetism,  and 
all  points  in  South  magnetic  latitude  have  North  or  red 
magnetism. 


THE  EARTH'S  MAGNETISM  57 

The  dip  increases  from  0°  at  the  magnetic  equator  to  90° 
at  the  magnetic  poles;  the  "total  force"  increases  from  a 
minimum  at  the  magnetic  equator  to  a  maximum  at  the 
"  magnetic  foci,"  of  which  there  are  two  in  each  hemisphere, 
located  about  in  52°  N.,  92°  W.;  and  70°  N.,  120°  E.;  70°  S., 
145°  E. ;  and  50°  S.,  130°  E.  The  total  force  at  the  foci  is 
between  two  and  three  times  that  at  the  magnetic  equator. 

The  magnetic  meridian. — The  magnetic  meridian  of  any 
place  is  that  great  vertical  circle  in  the  plane  of  which  the 
"  line  of  force  "  lies. 

The  variation. — Excepting  at  points  along  two  lines,  one 
at  present  passing  through  Brazil  and  the  eastern  part  of  the 
United  States,  and  the  other  through  Australia,  the  Arabian 
Sea,  and  the  Black  Sea,  called  "lines  of  no  variation,"  the 
magnetic  meridian  nowhere  corresponds  with  the  true  merid- 
ian, but  inclines  to  the  East  or  West  of  it,  making  with  it  at  a 
given  place  an  angle  called  the  "  variation  "  at  that  place.  As 
a  compass  needle  is  constrained  by  its  mode  of  suspension  to 
move  only  in  a  horizontal  plane,  variation  may  be  defined  as 
the  angle  through  which  the  compass  needle  is  deflected  from 
true  North  by  terrestrial  magnetism  alone. 

45  Elements  of  the  earth's  magnetism. — The  distribution 
of  the  earth's  magnetism  at  any  place  may  be  indicated  by  its 
three  elements : 

(a)  The  variation. 

(b)  The  dip. 

(c)  The  total  force  or  magnetic  intensity. 

(a)  and  (c)  are  found  by  means  of  the  magnetometer  and 
(b)  by  the  dip  circle,  but  for  the  purpose  of  representing  the 
amount  and  direction  of  the  earth's  force  on  the  needle,  in- 
stead of  considering  the  total  force  on  the  needle,  it  is  more 
convenient  to  consider  the  components  of  that  force,  viz. : 

(1)  The  horizontal  force,  or  that  component  in  the  direc- 
tion of  a  tangent  to  the  earth's  surface,  and  in  the  plane  of 
the  magnetic  meridian. 


58 


NAVIGATION 


(2)  The  vertical  force,  or  that  component  acting  down- 
wards and  at  right  angles  to  the  above. 

Relation  of  dip  and  the  forces  of  the  earth's  magnetism. — 
Letting  6  =  the  magnetic  dip, 
T  =  the  total  force, 
H  =  the  horizontal  force, 

Z  =  the  vertical  force,  we  shall  have  (Pig.  18), 
H—T  cos  0. 
Z  —  T  sin  6. 


Magnetic 


Meridian 


FIG.  18. 

46.  Charts  of  the  earth's  magnetic  elements. — The  TJ.  S. 

Hydrographic  Office  (also  corresponding  offices  in  foreign 
countries)  issue  variation  charts,  charts  of  magnetic  dip  and 
also  of  the  horizontal  intensity  of  the  earth's  magnetism.  The 
tide  tables,  issued  annually  by  the  U.  S.  C.  and  G.  Survey, 
give  the  variation  at  most  of  the  world's  seaports. 

The  variation  chart. — This  shows  by  lines  of  equal  value, 
called  isogonic  lines,  drawn  at  convenient  intervals,  the 
amount  and  direction  of  the  variation  over  the  surface  of  the 
globe.  Generally  speaking  the  variation  is  westerly  over  the 
Atlantic  Ocean,  the  Mediterranean,  and  the  Indian  Ocean  ex- 
cepting the  Bay  of  Bengal ;  easterly  over  the  Bay  of  Bengal, 
the  Pacific  Ocean,  the  Gulf  of  Mexico,  and  the  Caribbean  Sea. 

The  chart  of  magnetic  dip. — This  shows  by  lines  of  equal 


RELATION  OF  MERIDIANS  59 

value,  called  isoclinal  lines,  drawn  at  intervals  of  one  degree, 
the  magnetic  dip  over  the  surface  of  the  globe.  In  the  re- 
gions of  northerly  dip,  where  the  North  end  of  the  needle  is 
drawn  downward,  these  lines  are  full;  in  the  regions  of 
southerly  dip,  where  the  South  end  of  the  needle  is  drawn 
downwards,  they  are  broken  lines.  The  annual  rates  of  change 
of  dip  are  expressed  in  minutes  of  arc  by  numbers  in  the  re- 
gions where  they  are  placed ;  a  plus  sign  indicating  increasing, 
a  minus  sign  decreasing  dip. 

Taking  the  horizontal  force  at  the  magnetic  equator  as 
unity,  the  increase  of  dip  with  magnetic  latitude,  as  shown 
by  this  chart,  is  approximately  in  accordance  with  the  formula 
tan  dip  =  2  tan  mag.  Lat. 

The  chart  of  horizontal  force. — This  shows  the  horizontal 
intensity  expressed  in  C.  G.  S.  units  by  lines  of  equal  value. 

An  inspection  of  this  chart  will  show  that  the  horizontal 
force  is  a  maximum  near  the  magnetic  equator  and  dimin- 
ishes as  we  approach  the  magnetic  poles  where  it  is  zero. 
Since  the  horizontal  intensity  is  the  directive  force  on  the 
needle,  it  is  plain  that  a  disturbing  influence  would  have 
greater  effect  on  the  needle  when  the  value  of  H  is  less  and 
vice  versa ;  and,  therefore,  at  places  in  high  magnetic  latitudes, 
the  needle  is  less  reliable  than  at  places  near  the  magnetic 
equator,  when  subjected  to  the  same  influences  antagonistic 
to  that  of  the  earth. 

As  a  knowledge  of  the  value  of  H  in  the  locality  of  the 
ship  is  often  necessary  in  compass  work,  the  navigator  will 
find  this  chart  a  useful  one.  With  the  value  of  H  from  this 
chart  and  the  value  of  0  from  the  chart  of  magnetic  dip,  the 
value  of  Z ',  the  vertical  component  of  the  earth's  total  force 
at  a  place,  may  be  found.  Charts  of  Z  may  also  be  used. 

47.  Relation  of  true  and  magnetic  meridians. — From  what 
has  preceded,  it  is  plain  that  a  compass  needle,  constrained 
by  mechanical  arrangements  to  move  in  a  horizontal  plane 


60 


NAVIGATION 


and  installed  on  board  a  perfectly  stable  and  non-magnetic 
ship,  will  lie  in  the  magnetic  meridian,  at  an  angle  called  the 
variation  with  the  true  meridian,  to  the  eastward  or  westward 
of  it,  depending  on  the  geographical  position,  and  "will  pos- 
sess a  directive  force  depending  on  the  magnetic  latitude. 
The  variation  is  not  only  different  in  different  localities,  ex- 
cept at  those  places  on  the  same  isogonic  line,  but  it  is  differ- 
ent in  the  same  locality  at  different  times,  owing  to  a  small 
but  gradual  and  constant  motion  of  the  magnetic  poles. 

The  "  line  of  no  variation/'  which  now  passes  through  the 
Arabian  and  Black  seas,  was  a  little  to  the  westward  of  the 


PIG.  19. 


FIG.  20. 


Azores  in  1492,  and  it  is  recorded  that  Columbus,  on  his 
westward  voyage,  noted  the  change  in  the  compass  bearing  of 
the  pole  star;  the  needle  at  first  pointing  to  the  eastward  of 
the  pole  star,  then  directly  at  it,  and  finally  to  the  westward 
of  it  as  the  voyage  progressed. 

Besides  the  regular  and  periodic  annual  change,  the  needle 
is  subject  to  a  slight  diurnal  change,  moving  gradually  back 
and  forth  through  a  very  small  arc. 

Variation  is  shown  on  all  navigational  charts  at  the  com- 
pass rose  and  also  by  isogonic  lines. 

All  magnetic  courses  and  bearings  are  estimated  from  the 
magnetic  meridian,  but  it  is  often  desirable  and  necessary  to 
find  the  corresponding  angles  from  the  true  meridian,  which 


EULES  FOR  VARIATION  61 

may  be  found  by  applying  the  variation  of  the  place  to  the 
magnetic  course  or  bearing  in  the  proper  way. 

In  Figs.  19  and  20,  let  ON  be  the  true  meridian;  OM  the 
magnetic  meridian ;  V  =  NOM  the  variation  of  the  place ;  OH 
the  keel  line  or  direction  of  the  ship's  head ;  then  NOH  is  the 
true  course  and  MOH  the  magnetic  course,  considered  positive 
to  the  right.  In  Fig.  19  the  North  end  of  the  needle  is  drawn 
to  the  eastward  and 

The  true  course  =  the  magnetic  course  -f-  variation. 
In  Fig.  20  the  North  end  of  the  needle  is  drawn  to  the  west- 
ward and 

The  true  course  =  the  magnetic  course  —  variation. 

48.  Rule  for  naming  variation. — Mark  it  East  (E)  or  +,  if 
the  North  end  of  the  needle  is  drawn  to  the  right,  the  observer 
considered  as  at  the  center  of  the  compass  and  looking  in  the 
direction  of  that  end  of  the  needle;  mark  it  West  (W)  or  — , 
if  the  North  end  of  the  needle  is  drawn  to  the  left,  observer 
as  before. 

When  the  North  point  of  the  needle  is  drawn  to  the  right, 
the  magnetic  meridian  is  to  the  right  of  the  true  meridian, 
and  the  magnetic  bearing  of  a  fixed  object  is  to  the  left  of  its 
true  bearing  by  the  amount  of  the  variation;  similarly  when 
the  North  point  of  the  needle  is  drawn  to  the  left,  the  mag- 
netic meridian  is  to  the  left  of  the  true  meridian,  and  the 
magnetic  bearing  of  a  fixed  object  is  to  the  right  of  its  true 
bearing  by  the  amount  of  the  variation. 

Rule  for  applying  variation. — Hence,  when  applying  varia- 
tion to  magnetic  courses  or  bearings  to  obtain  the  correspond- 
ing true  courses  or  bearings,  looking  from  the  center  of  the 
compass  toward  the  compass  rhumb,  apply  variation  to  the 
right  when  E.  or  + ;  to  the  left  when  W.  or  — .  Or,  if  the 
magnetic  course  or  bearing  is  in  degrees,  add  the  variation  if 
E.  or  +,  subtract  it  if  W.  or  — .  To  find  the  magnetic 
courses  (or  bearings)  from  the  true  courses  (or  bearings)  do 
the  reverse. 


62  NAVIGATION 

EXAMPLES,  VARIATION  EAST. 

49.  Given  the  following-  magnetic  courses,  to  find  the  cor- 
responding true  courses : 

Magnetic  Course  .         85°  137°  230°  353° 

Variation    +15  +15  +15  +15 


True  Course 100°  152°  245°  8° 

EXAMPLES,  VARIATION  WEST. 
Given  magnetic  courses,  to  find  the  true  courses: 

Magnetic  Course  .         85°  137°  230°  3° 

Variation    —15  —15  —15  —15 

True  Course 70°  122°  215°  348° 

50.  local  attraction. — There  is  cause  of  disturbance  of  the 
compass  needle,  when  the  ship  is  in  certain  localities,  due  to 
the  fact  that  the  mineral  substances  in  the  land  under  the 
water  possess  magnetic  properties,  especially  in  shallow  waters 
of   volcanic   regions.     Well-authenticated   observations    show 
that  the  navigator  must  be  on  his  guard  against  the  dangers 
of  this  attraction  on  the  coasts  of  Iceland,  off  Cape  St.  Francis, 
in  Odessa  Bay,  off  the  coast  of  Madagascar,  off  the  volcanic 
islands  near  Java,  at  the  Isles  de  Los,  and  especially  near 
Cossack  in  North  Australia. 

51.  Deviation. — So  far  we  have  considered  the  compass  as 
if  installed  on  board  an  absolutely  non-magnetic  ship,  and  as 
affected  only  by  terrestrial  magnetic  influences,  with  variation 
as  its  only  correction  or  error.     However,  when  that  same 
compass  is  mounted  on  board  an  iron  or  steel  ship,  it  is  sub- 
ject to  further  error  in  its  indications.     Besides  having  a 
directive  force  in  the  magnetic  meridian  given  it  by  H,  the 
earth's  horizontal  intensity,  the  North  end  of  the  needle  is 
acted  upon  by  the  general  magnetism  induced  in  the  iron  or 


RULES  FOR  DEVIATION  63 

steel  of  the  ship  by  the  earth's  inducing  forces,  with  the  result 
that  the  needle  assumes  a  resultant  direction,  the  angle  be- 
tween which  and  the  magnetic  meridian  is  called  "  deviation/' 

When  the  deviation  is  zero,  the  ship's  force  acts  in  the 
magnetic  meridian,  increasing  or  diminishing  the  earth's  di- 
rective force. 

The  deviation  of  a  compass  varies  with  the  ship  in  which 
it  is  mounted ;  varies  according  to  position  in  the  same  ship ; 
and  for  a  given  position,  under  like  circumstances,  varies  in 
amount  and  direction  according  to  the  heading  of  the  ship. 
It  also  varies  with  change  of  ship's  position  on  the  earth's 
surface.  For  these  reasons  the  deviations  of  every  compass, 
mounted  on  board  and  used  for  navigating  or  steering  the 
ship,  should  be  determined  for  every  15°  compass  rhumb  at 
a  time  when  the  vessel  is  on  an  even  keel,  in  her  normal  sea- 
going condition,  with  projectiles,  guns,  davits,  cranes,  re- 
movable masses  of  iron,  etc.,  secured  as  if  for  sea.  • 

The  deviations  of  the  standard  compass,  which  alone  must 
be  used  for  navigating  the  ship,  should  be  tabulated  and  a 
corrected  copy  of  the  table  should  be  kept  on  deck  for  the 
use  of  the  officer  of  the  deck  and  the  navigator.  Such  a  table 
is  needed  for  finding  the  magnetic  course  from  the  compass 
course  steered;  for  correcting  the  compass  bearings  of  fixed 
objects  on  shore;  or  for  obtaining  the  compass  course  to  be 
steered  to  make  good  a  certain  magnetic  course. 

52.  Rule  for  naming  deviation. — Mark  it  East  (E)  or  + 
if,  under  the  influence  of  the  ship's  magnetism,  the  North 
end  of  the  needle  is  drawn  to  the  eastward,  or  to  the  right  of 
the  magnetic  meridian;  mark  it  West  (W)  or  —  if  the  North 
end  of  the  needle  is  drawn  to  the  westward,  or  to  the  left  of 
the  magnetic  meridian. 

When  the  North  point  of  the  needle  is  drawn  to  the  right, 
the  observer  at  the  center  and  looking  in  the  direction  of  that 
point,  the  compass  meridian  is  to  the  right  of  the  magnetic 


64  NAVIGATION 

meridian  and  the  compass  bearing  of  a  fixed  object  is  to  the 
left  of  its  magnetic  bearing  by  the  amount  of  the  deviation; 
similarly  when  the  North  point  of  the  needle  is  drawn  to  the 
left,  the  compass  meridian  is  to  the  left  of  the  magnetic  merid- 
ian, and  the  compass  bearing  of  a  fixed  object  is  to  the  right 
of  its  magnetic  bearing  by  the  amount  of  the  deviation. 

Kule  for  applying  deviation. — When  applying  deviation  to 
compass  courses,  or  bearings,  to  obtain  magnetic  courses,  or 
bearings,  looking  as  if  from  the  center  of  the  compass  toward 
the  compass  rhumb,  apply  the  deviation,  due  to  the  ship's 
heading  at  the  time,  to  the  right  when  E.  or  -\- ;  to  the  left 
when  W.  or  — .  Or,  if  the  compass  course  or  bearing  is  in 
degrees,  add  the  deviation  if  E.  or  -)-?  subtract  it  if  W.  or  — . 
To  find  the  compass  course  from  a  given  magnetic  course  do 
the  reverse. 

In  this  connection,  attention  is  particularly  called  to  the 
fact  that  all  bearings  are  to  be  corrected  for  the  deviation  due 
to  the  direction  of  the  ship's  head  at  the  moment  they  were 
taken. 

53.  Compass  error. — When  variation  and  deviation  are  to 
be  allowed  for  at  one  time,  add  them  algebraically,  giving  the 
name  of  the  greater  to  the  result  which  is  known  as  "  compass 
error,"  generally  written  C.  E.  To  obtain  the  true  course  or 
bearing  from  a  given  compass  course  or  bearing,  apply  the 
C.  E.  to  right  if  E.  or  + ;  to  the  left  if  W.  or  — ,  looking 
from  the  center  of  the  compass  toward  the  compass  rhumb. 
Or,  if  the  compass  course  or  bearing  is  in  degrees,  add  the 
compass  error  if  E.  or  -(-,  subtract  it  if  W.  or  — .  To  obtain 
a  compass  course  from  a  given  true  course  do  the  reverse. 

For  office  work  and  in  examples  similar  to  4,  5,  6,  and  8  the 
signs  -f-  and  —  are'  preferable  to  the  terms  E.  and  W.  The 
use  of  the  latter,  however,  will  be  illustrated  in  examples 
1,  2,  3,  and  7. 

Example  1.— Given  Var.  =  13°  W,  Dev.  on  North  (p.  c.) 


COURSE  CORRECTIONS  65 

2°  E,  on  NE  (p.  c.)  1°  W,  and  on  East  (p.  c.)  4°  W;  find 
the  true  courses  corresponding  to  the  above  compass  courses. 


Var.  13°  W  Course  (p.  c.) 

North 

Dev.  2°  E       C.  E.  11°  W 

C. E.  11°  W 

True  Course          349° 


Var.  13°  W  Course  (p.  c.) 

45° 
Dev.  1°  W      C.  B.  14°  W 


C.  E.  14°  W  j 

True  Course  31° 


Var.  13°  W  Course  (p.  o). 
Dev.  4°W       C.  E.17°W 

C.  E.1,°W 

True  Course  73° 


54.  Leeway. — With  sailing  ships,  the  wind,  besides  driving 
the  ship  in  the  direction  of  her  keel,  frequently  forces  her 
bodily  to  leeward,  so  that  the  course  through  the  water  is  to 
leeward  of  the  one  steered. 

This  angle  between  the  course  and  the  direction  the  ship  is 
actually  moving,  as  indicated  by  the  ship's  wake,  is  the  leeway. 
Being  always  from  the  wind,  as  a  correction  it  is  marked  East 
when  the  ship  is  on  the  port  tack,  West  when  the  ship  is  on 
the  starboard  tack.  Here  also,  East  is  +  and  West  is  — . 

Given  a  course  (p.  c.),  to  find  the  true  course. — 

Ex. '2.— A  schooner  sails  28°  (p.  s.  c.),  Dev.  6°  E,  Var. 
from  chart  21°  E,  wind  SE,  leeway  11°.  Find  the  true 
course. 

Var.  21°  E.  Course  (p.  c.)  28° 

Dev.  6     E.  Correction  16    E. 

Leeway         11    W.  True  course  44° 

Correction   16°  E. 

Given  the  true  course,  to  find  the  compass  course. — 

Ex.  3. — The  true  course  to  destination  from  the  ship's  posi- 
tion is  22°  30',  Var.  15°  W,  Dev.  6°  E.  The  ship  will  be  on 
the  port  tack,  probable  leeway  6°.  Eind  the  course  to  be 
steered. 

Var.  15°  W.  True  course  22°  30' 

Dev.  6     E.  Reversed  correction        3      E. 

Leeway          6     E.  Compass  course  25°  30' 

Correction      3°  W. 


66  NAVIGATION 

The  correction  in  this  example  being  3°  W  is  applied  the 
reverse  way,  or,  as  if  it  is  easterly. 

The  word  correction  is  used  here  because  leeway  is  not  an 
error  of  the  compass.  Strictly  speaking  variation  is  not  an 
error  and  cannot  be  compensated  for;  deviation  only  is  an 
error. 

SECTION  III. 

55.  Finding  the  deviation. — For  reasons  that  are  now  ap- 
parent it  is  essential  that  a  table  of  deviations  should  be  ob- 
tained for  all  compasses  mounted  on  board  as  soon  as  possible 
after  a  vessel  is  commissioned,  that  the  table  for  the  standard 
compass  should  be  checked  from  day  to  day,  and  a  new  one 
made  out  after  any  marked  change  of  magnetic  latitude. 

For  a  new  vessel  built  of  iron  or  steel,  observations  should 
be  made  on  the  24  equidistant  15°  rhumbs  before  compen- 
sation; after  compensation  the  residual  deviations  may  be 
found  by  observing  on  12  equidistant  headings,  though  in  both 
cases,  if  possible  to  do  so,  it  would  be  better  to  swing  with  both 
helms  and  to  take  the  mean  of  the  two  deviations  on  each 
heading  as  the  correct  deviation  for  that  heading. 

As  the  ship  is  steadied  on  each  heading  and  observations 
for  deviation  are  made  at  the  standard,  the  ship's  head  should 
be  noted  by  observers  at  the  steering  and  pilot-house  com- 
passes ;  then  from  the  headings  and  deviations  of  the  standard, 
the  magnetic  heading  of  the  ship  at  each  observation  may  be 
found.  A  comparison  of  each  magnetic  heading  with  the  cor- 
responding heading  by  each  of  the  compasses  will  give  the 
deviation  for  the  heading  of  the  compass  compared.  Before 
an  observation  is  taken  on  any  heading,  the  ship  should  be 
steadied  on  it  for  three  or  'four  minutes,  in  order  that  the 
needle  may  be  at  rest  and  under  magnetic  influences  normal 
for  that  heading  at  the  time  of  observation. 

The  ship  itself  should  be  steady,  or  its  motion  a  minimum, 
when  the  observer  takes  his  observations. 


EECIPROCAL  BEARINGS  67 

The  deviation  may  be  obtained  by  any  one  of  four  methods : 

(1)  By  reciprocal  bearings; 

(2)  By  bearings  of  a  distant  object; 

(3)  By  azimuths  or  amplitudes  of  a  celestial  body; 

(4)  By  ranges  of  known  magnetic  bearing;  or,  by  two  or 
more  of  the  above  combined. 

56.  (1)  By  reciprocal  bearings. — This  method  is  available 
when  the  ship  is  in  a  basin  or  a  smooth  harbor,  and  the  com- 
passes are  free  from  all  disturbing  influences  except  the  ship's 
own  magnetism  and  that  of  the  earth;  and  when  there  is  a 
suitable  position  on  shore  for  mounting  a  compass  where  there 
are  no  local  magnetic  influences,  above  or  below  ground,  to 
disturb  its  readings. 

A  careful  observer  is  sent  ashore  with  a  spare  compass  on 
a  tripod  which  is  placed  where  it  can  be  seen  distinctly  from 
the  ship  with  the  naked  eye,  in  a  spot  absolutely  free  from 
all  local  magnetism. 

The  requisite  warps  having  been  prepared,  the  ship  is 
swung  around  so  as  to  bring  her  head,  per  standard  compass, 
upon  each  heading  on  which  observations  for  deviation  are  to 
be  taken;  of  course,  if  circumstances  permit,  it  is  advisable 
to  observe  on  each  of  the  24  equidistant  15°  rhumbs. 

Then,  by  means  of  prearranged  signals,  the  mutual  bearings 
of  this  shore  compass  and  the  standard  compass  on  board  are 
observed  at  the  moment  when  the  ship's  head  is  steady,  and 
has  been  steady  at  least  three  minutes,  on  each  of  the  required 
compass  headings.  To  guard  against  mistakes,  the  time 
of  each  bearing  should  be  observed,  both  on  board  and  ashore, 
by  compared  watches;  and  it  is  advisable  for  the  shore  ob- 
server to  mark  the  time  and  bearing  of  the  standard  from 
the  shore  compass  at  each  observation  on  a  blackboard  pro- 
vided for  the  purpose,  so  that  in  case  of  an  apparent  incon- 
sistency, the  observations  can  be  immediately  repeated  and 
the  necessity  obviated  for  again  swinging  the  ship. 

NOTE.— Whenever  bearings  are  taken  with  the  azimuth  circle,  it  should  be  hori- 
zontal with  the  bubble  of  the  level  centered.  Celestial  bodies  should  be  observed 
for  deviation  when  on  or  near  the  P.  V.  and  at  a  low  altitude  (see  Art.  222). 


68  NAVIGATION 

The  bearing  of  the  standard  from  the  shore  compass  at  a 
given  instant,  reversed,  is  the  correct  magnetic  bearing  of  the 
shore  compass  from  the  standard  at  that  instant,  and  the 
difference  between  this  magnetic  bearing  and  the  bearing  taken 
at  the  standard  on  board  at  the  same  time  will  be  the  devia- 
tion due  to  the  particular  heading  of  the  ship  at  the  moment 
of  observation. 

This  deviation  is  marked  according  to  the  rule  given  in 
Article  52. 

The  results  of  the  swinging  are  recorded  as  in  the  form 
used  in  the  following  example  solved  on  page  69. 

Ex.  4- — Having  swung  a  Monitor  for  deviations  of  the 
standard  and  battle  compasses  by  method  of  reciprocal  bear- 
ings, find  the  deviations  of  standard  on  24  rhumbs,  and  of 
battle  compass  for  the  magnetic  headings.  Data  as  in  form. 

57.  (2)  By  bearings  of  a  distant  terrestrial  object. — This 
method  is  convenient  when  the  ship  is  at  anchor  in  a  harbor, 
or  roadstead,  with  the  object  so  far  distant  that  the  magnetic 
bearing  will  not  alter  sensibly  as  the  ship  heads  on  the  various 
headings — say  about  eight  to  ten  miles  for  a  ship  swinging, 
anchored  at  short  stay.  This  method  may  be  used  at  sea,  the 
ship  steaming  around  an  entire  circle,  provided  the  object  is 
so  far  distant  that  the  parallax  does  not  exceed  30',  the  paral- 
lax being  the  angle  whose  tangent  equals  the  radius  of  the 
circle  in  which  the  ship  is  swinging  divided  by  the  mean  dis- 
tance of  the  object.  At  sea,  even  under  the  most  favorable 
conditions,  it  involves  more  or  less  error;  and,  if  the  ship  is 
in  the  locality  of  tides  or  currents,  this  method  should  not  be 
used  with  the  ship  underway. 

By  this  method,  a  distant  but  distinctly  visible  object,  as  a 
clearly  defined  point  of  a  distant  peak,  a  light-house,  or  other 
mark,  is  observed  as  the  ship  at  anchor  swings  slowly  to  tide, 
is  steamed  around,  or  swung  at  her  moorings,  but  steadied 
sufficiently  long  on  each  heading  to  allow  the  magnetism  of 
the  ship  to  settle  down. 


EECIPROCAL  BEARINGS 


69 


«: 

«« 


CO 


T- 
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|  |  |  |  | 


OOOOO 
OTHCOO(M 


goo 

O 

CO 


OOOtfiOOOOlOOlOoOOOOlO 


OlO 
OrH 


iHTHTHrHi-HTH<M<M<MW<MCOCOOOWCO 


21 

si 


•}<MCJ<M(M<?3SQCJCOCO 


§   -2 
%+<• 


^a  o 
°6 


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OlOOOlOOOlOOOOOOOO»O 


5133 

CQ  §,020 

O 


<M<M<M<M(M<?1<M<N<M 


i— <c^Ttiiot-o6o»-4CO 

(MWIMWWWCOCOCO 


70 


NAVIGATION 


The  difference  between  these  bearings  and  the  magnetic 
bearing  of  the  distant  object,  or,  in  other  words,  the  compass 
bearing  it  would  have  from  on  board  if  the  compass  were  not 
disturbed  by  the  attraction  of  the  iron  of  the  "ship,  will  be 
the  deviation  of  the  compass  for  the  headings  of  the  ship  at  the 
lime  the  bearings  were  taken,  marked  as  per  rule  in  Article  52. 

The  magnetic  bearing  of  the  distant  object  may  be  found : 

(a)  From  a  chart,  if  it  is  one  of  a  recent  and  reliable  sur- 
vey. 

(b)  From  the  mean  of  four  or  more  compass  bearings  on 
equidistant  compass  headings. 

(c)  From  a  true  or  astronomical  bearing,   applying  the 
known  variation  of  the  place. 

Ex.  5. — As  a  ship  swung  to  tide,  at  anchor,  the  follow- 
ing bearings  were  taken  of  the  sharp  peak  of  a  distant  moun- 
tain by  the  standard  compass.  Eequired  the  deviations  on 
the  headings  indicated. 


Head 

(p.  s.  c.). 

Bearings 
of  object 
(p.  8.  c.). 

Deviations 
of 
Standard. 

Head 
(p.  s.  c.). 

Bearings 
of  object 

(P.S.C.). 

Deviations 
of 
Standard. 

0° 

222° 

+  3°  22V 

180° 

227° 

—  1°  37V 

45 

221 

+  4   22% 

225 

230 

-4   37*2 

90 

219 

+  6   22a/2 

270 

232 

-6    37V2 

135 

223 

+  2   22V2 

315 

229 

—  3    37^ 

=  225 


The   correct   magnetic   bearing,   taken   as 
the  mean  of  all  eight  bearings  (p.  c.), 

58.  (3)  By  bearings  of  a  celestial  body. — This  method 
may  be  used  whenever  either  of  the  previous  methods  is 
available,  provided  the  body  observed  is  not  too  high  in  alti- 
tude. It  is  particularly  important  as  it  is  the  only  one  avail- 
able at  sea. 

In  this  method  the  ship  is  steadied  for  the  required  time 
on  each  heading  per  standard  compass.  By  this  compass  the 

NOTE.— The  magnetic  bearing  as  obtained  above  under  (&)  will  include  any  value 
of  the  constant  deviation  A  that  may  exist  for  the  particular  compass  and  location. 


TIME  AZIMUTHS  71 

bearing  of  the  celestial  body  is  observed  when  the  ship  is 
steady  on  each  heading,  at  the  same  moment  the  headings  by 
the  steering  and  other  compasses  are  read  and  the  time  noted 
by  a  chronometer,  or  a  watch  compared  with  a  chronometer 
whose  error  on  G.  M.  T.  is  known.  From  the  longitude  and 
times  the  hour  angles  are  found;  then  with  the  latitude,  the 
body's  declination,  and  the  hour  angle  at  the  time  a  com- 
pass bearing  was  taken,  the  corresponding  true  bearing  may 
be  found  in  the  azimuth  tables.  The  difference  between 
the  true  and  the  compass  bearing  of  the  body  at  the  same 
instant  is  the  compass  error  from  which  the  deviation  may 
be  found  by  applying  the  variation  from  the  chart.  If  the 
compass  errors  are  obtained  on  equidistant  headings,  and  the 
iron  about  the  compass  is  symmetrical,  the  mean  of  the  com- 
pass errors  will  give  the  variation.  If  the  sun  is  the  body  ob- 
served, the  true  azimuth  will  be  found  in  the  azimuth  tables ; 
using  latitude,  declination,  and  local  apparent  time  as  argu- 
ments. 

Ex.  6. — April  7,  1905,  a  U.  S.  torpedo-boat  destroyer 
was  swung  for  deviations  of  the  standard  compass,  all  cor- 
rectors having  been  removed.  Data  as  given  in  the  form 
for  a  Compass  Eeport  on  page  73.  L  39°  K,  X  76°  24'  W. 

The  watch  time  of  observation  and  the  compass  bearing  of 
the  sun  are  recorded,  respectively,  in  columns  2  and  5,  oppo- 
site the  proper  heading  per  compass. 

The  error  of  the  watch  on  L.  A.  T.  is  found  from  the  data 
under  caption  "  Local  apparent  time  by  chronometer  "  and  is 
entered  in  column  3.  This  applied  to  the  watch  times  in 
column  2  will  give  the  corresponding  local  apparent  times  of 
observation  recorded  in  column  4. 

The  work  of  finding  the  sun's  true  bearing  from  the  azi- 
muth tables  may  be  facilitated,  without  loss  of  material  accu^ 
racy,  by  computing  a  constant  for  the  minutes  of  declination 
and  latitude  corresponding  to  the  approximate  middle  instant 


72  NAVIGATION 

of  observation;  the  declination  used  being  taken  from  the 
Nautical  Almanac  for  the  Greenwich  time  of  the  middle 
instant.  The  mean  declination  thus  found  in  this  example  is 
6°  42'  +  N  =  6°. 7  N  and  the  middle  L.  A.  T.  is  8h  44m  10s. 
The  constant  is  thus  found  for  minutes  of  latitude  and  declin- 
ation, taking  the  L.  A.  T.  equal  to  8h  40m. 

Dec      fi°  N         on  pa£e  90  of  azimuth  tables. 


For  Dec.  7°  N  Z  —  N  111    59  E 

Diff.  for  1°  of  Dec.    —       (_)  52'  For  minutes  of  Dec.  (—)  36' 

Diff.  for  0°. 7  of  Dec.  —       (— )  36  For  minutes  of  Lat.    r=      0 

Constant    .  .    (  — )  36'. 


Lat.  40°  N    I 
Dec.     6°N     L°nJ 
A.  T.    SMO^J  *—* 


page  92. 

113°  29'  E 


For  Lat.  39°  NZ— N  112    51  E 
Diff.  for  1°  Lat.    —   (+)  0  38' 
DiffforO°.OLat.  —  0' 

Then,  using  page  90  of  the  Azimuth  Tables,  with  Lat.  39° 
N  and  Dec.  6°  N",  find  the  true  azimuth  interpolated  for  the 
local  apparent  time  of  column  4;  apply  the  constant  and 
enter  result  in  column  6. 

The  difference  between  columns  5  and  6,  marked  in  accord- 
ance with  the  rules  of  Article  53,  will  be  the  compass -error 
on  the  heading  opposite  in  column  1,  and  will  be  entered  in 
column  7. 

The  difference  between  the  mean  of  equidistant  azimuths 
in  columns  5  and  6  is  the  variation  -f-  constant  A,  as  shown 
in  column  8.  The  mean  of  the  compass  errors  on  equidistant 
headings  should  give  the  same  result,  variation  +  constant. 
A,  if  no  mistakes  have  been  made.  If  accurately  known, 
the  value  of  A  should  be  applied  to  the  above-mentioned  dif- 
ference to  obtain  the  variation  by  observation. 

The  algebraic  difference  between  columns  7  and  8,  that  is 
the  compass  error  and  the  variation,  is  the  deviation  to  be 

NOTE. — Before  being  entered  in  column  6,  the  true  azimuth  by  tables  should  be  expressed, 
like  the  compass  azimuth,  in  the  form  of  Z.\  which  is  the  azimuth  measured  from  North,  around 
to  the  right,  from  0°  to  360°. 


TIME  AZIMUTHS 


73 


eviations  o 
Compass. 
No.  7203. 


I 


?|M 


Azimuth  of 
sun  by  Stand 
ard  Compass 
No.  7*03. 


l  appar- 
times 
of 
rvation. 


O     . 


CL 


08  9—  = 

g[  —  = 

8?  9—= 

ff  81  i  =  ' 


'uoi^BAJtasqo  Xq  uo^'Bia'B^ 

oj'uoqAi^idd'B)  y^u^sno3 

+  uoi^BAJtasqo  ^q  uot^Bta'BA 


MM 


s^sns 


t-  t-t-GOOOCOOOCOOOQOCOOOaOOOCiO5OSO5OiOSO5O5OiO> 


'9ca[(; 


peppB  eq 
no 


'8  OS  'va  g 
jo 


Times  of 
observation 
by  watch. 


a'cOl^-i—  (t—  ^H'!O^Ht'- 
XS^^iOOOi^f^ 


and 
tion 


on  of  compass 
lars  of  compe 


74  NAVIGATION 

entered  in  column  9.  Bearing  in  mind  the  fact  that  the  + 
sign  is  given  to  easterly  errors,  variation,  and  deviation,  and 
the  —  sign  to  westerly  errors,  variation,  and  deviation,  and 
also  that  deviation  equals  compass  error — variation,  the  sign 
of  the  deviation  should  be  apparent.  Easterly  deviation  may 
be  marked  either  -f-  or  E.,  westerly  deviation  ( — )  or  W. 

The  magnetic  azimuth  of  the  ship's  head  may  be  found  by 
applying  the  deviations  of  the  standard  compass  to  the  head- 
ings per  standard. 

At  the  instant  of  observing  the  sun's  azimuth  per  standard, 
the  ship's  head  by  all  other  compasses  on  board  should  be 
noted.  The  readings  of  these  compasses  compared  with  the 
corresponding  magnetic  azimuths  of  the  ship's  head  will  give 
the  deviations  of  the  compasses  on  their  particular  headings, 
the  deviations  being  marked  as  per  rule  Article  52. 

59.  (4)  By  ranges. — Eanges,  whose  magnetic  bearings  are 
known,  may  be  found  in  various  localities,  having  been  speci- 
ally laid  out,  or  formed  under  natural  conditions.     The  data 
concerning  a  number  of  such  ranges  have  been  published  in  a 
pamphlet  by  the  U.  S.  C.  and  G-.  Survey. 

When  steaming  across  these  ranges  on  various  headings, 
the  compass  bearing  of  the  range  may  be  taken. 

The  deviation  for  any  heading  will  be  the  difference  be- 
tween the  compass  bearing  of  the  range  on  that  heading  and 
the  known  magnetic  bearing,  marked  easterly  when  the  mag- 
netic bearing  is  to  the  right  of  the  compass  bearing,  westerly 
when  the  magnetic  bearing  is  to  the  left  of  the  compass 
bearing. 

60.  Napier's  diagram. — This  is  a  graphic  representation  of 
deviations  on  either  compass  or  magnetic  headings,  and  it 
furnishes  a  ready  method  of  finding  the  magnetic  course  cor- 
responding to  a  given  compass  course,  and  vice  versa. 

It  consists  of  a  vertical  line  of  convenient  length  divided 
into  24  equal  parts  representing  the  24  15°-  rhumbs  of  the 


A  GRAPHIC  METHOD  75 

Compass  courses  on  dotted  lines.  Magnetic  courses  on  solid  lines,. 


FROM  0°  NORTH  TO  1OO°  SOUTH 


DEVIATION 


FROM  1OO"  SOUTH  TO  36O°  NORTH 


DEVIATION 


DEVIATION 

EAST 


Curve  of  Deviations,  Napier's  Diagram. 


76  NAVIGATION 

compass;  beginning  at  the  top,  these  are  numbered  in  order, 
0°,  15°,  30°,  45°  up  to  360°  from  North  around  to  the  right. 
The  line  is  also  divided  into  360  equal  parts  representing  de- 
grees, numbered  at  every  fifth  degree  around  to  the  right. 
Usually  the  curve  is  shown  in  two  parts  as  on  page  75. 

The  vertical  line  is  intersected  at  15°  rhumbs  by  two  lines, 
a  plain  line  inclined  upward  and  to  the  right,  a  dotted  line 
inclined  upward  and  to  the  left,  each  making  an  angle  of  60° 
with  the  vertical. 

To  construct  a  curve. — Take  on  the  vertical  line  the  com- 
pass course  for  which  the  deviation  has  been  obtained ;  lay  off 
this  deviation,  to  the  scale  of  the  vertical  line,  on  the  dotted 
line  which  passes  through  the  course,  or  in  a  direction  paral- 
lel to  the  dotted  lines,  to  the  right  if  the  observed  deviation 
is  easterly,  to  the  left  if  westerly,  and  mark  the  point  so  ob- 
tained with  a  dot.  Having  done  this  for  each  observed  devia- 
tion, trace  a  fair  curve  through  the  points,  and  this  will  be 
the  deviation  curve.  The  deviations  should  be  obtained  on 
eight  or  more  rhumbs  equally  distributed  around  the  compass, 
but  these  need  not  be  15°  rhumbs.  If  not  possible  to  get 
the  deviations  on  more  than  four  rhumbs,  these  should  be  as 
near  the  quadrantal  points  as  possible. 

Rule  I.  From  a  given  compass  course  to  find  the  corre- 
sponding magnetic  course. 

Take  the  compass  course  on  the  vertical  line;  move  thence 
on  or  parallel  to  the  dotted  lines  till  the  curve  is  intersected, 
thence  on  or  parallel  to  the  plain  lines  till  the  vertical  line  is 
intersected.  This  point  in  the  vertical  line  will  be  the  re- 
quired magnetic  course. 

Rule  II.  From  a  given  magnetic  course  to  find  the  corre- 
sponding compass  course. 

Find  the  magnetic  course  on  the  vertical  line;  from  this 
point  move  on  or  parallel  to  the  plain  lines  till  the  curve  is 
intersected,  thence  on  or  parallel  to  the  dotted  lines  meeting 
the  vertical  line  in  the  required  compass  course. 


COMPASS  DEVIATIONS  77 

SECTION  IV. 
THE  THEORY  OF  COMPASS  DEVIATIONS. 

61.  Soft  iron  and  hard  iron. — Preliminary  to  investigating 
the  causes  of  deviations,  it  is  essential  to  consider  the  char- 
acter of  the  iron  used  in  building  a  ship  and  the  influence  of 
the  earth's  magnetism  on  that  iron. 

Considering  its  physical  characteristics,  iron  may  be  desig- 
nated magnetically  by  the  terms  "  soft "  and  "  hard." 

Soft  iron  is  iron  which,  under  the  influence  of  a  magnetic 
force,  will  instantly  acquire  magnetism  by  induction,  but  will 
as  quickly  lose  it  when  that  force  is  removed.  In  other  words 
its  magnetism  is  transient  induced  magnetism. 

Hard  iron  is  less  susceptible  to  magnetic  induction,  but, 
when  once  magnetized,  it  retains  a  large  part  of  its  mag- 
netism permanently.  Furthermore,  the  greater  the  hardness 
and  the  less  easily  it  can  be  magnetized,  the  greater  the 
amount  of  magnetism  it  is  capable  of  retaining  and  the  longer 
it  will  retain  it. 

62.  Effect  of  earth's  magnetism  on  a  soft  iron  rod. — If  a 
rod  of  soft  iron  be  held  in  the  direction  of  the  "  line  of  force," 
it  will  instantly  become  magnetic.     If  in  North  latitude, 
the  lower  end  will  have  induced  in  it  North  or  red  magnetism, 
and  will  repel  the  North  end  of  the  compass  needle;  and  the 
upper  end  will  have  induced  in  it  South  or  blue  magnetism, 
and  will  attract  the  North  end  of  the  compass  needle. 

If  inclined  to  the  "line  of  force,"  its  induced  magnetism 
will  be  proportional  to  the  cosine  of  the  angle  of  inclination; 
therefore,  at  90°,  or  at  right  angles  with  the  "line  of  force," 
the  rod  will  be  in  a  neutral  condition;  beyond  90°  the  mag- 
netism will  be  reversed,  that  end  which  was  at  first  of  North 
polarity  will  have  South  polarity,  and  the  intensity  of  the 
magnetism  induced  will  increase  till  at  180°  it  will  again 
be  a  maximum. 


78  NAVIGATION 

If  instead  of  being  held  in  the  direction  of  the  "line  of 
force,"  the  rod  is  moved  in  a  horizontal  plane,  it  will  be  sub- 
ject to  induction  only  by  the  horizontal  component  of  the 
total  force,  so  that  if  placed  East  and  West  magnetic,  the 
bar,  being  at  right  angles  to  the  inducing  force,  will  be  in  a 
neutral  condition  and  have  no  effect  on  either  end  of  the  com- 
pass needle. 

If  held  in  any  other  position  in  the  horizontal  plane,  its 
South  end  will  attract,  and  its  North  end  will  repel  the  North 
end  of  the  compass  needle  with  a  force  proportional  to  the 
horizontal  intensity  multiplied  by  the  cosine  of  the  rod's  mag- 
netic azimuth. 

If  a  soft  iron  rod  be  held  in  a  vertical  position,  it  will  have 
magnetism  induced  in  it  by  the  vertical  component  of  the 
earth's  magnetism.  In  North  latitude,  the  upper  or  South 
end  will  attract  the  North  end  of  the  compass  needle;  at  the 
magnetic  equator,  being  perpendicular  to  the  line  of  force,  it 
will  be  in  a  neutral  condition;  and  in  South  magnetic  lati- 
tude the  polarity  will  be  reversed,  the  upper  end  then  having 
North  polarity  will  repel  the  North  end  of  the  needle. 

If  a  rod,  held  in  a  position  favorable  to  induction,  is  ham- 
mered, twisted,  bent,  or  otherwise  subjected,  to  mechanical 
violence,  the  amount  of  magnetism  it  will  receive  is  increased. 
This  magnetism  diminishes  more  or  less  rapidly  in  the  first 
few  weeks  but  a  portion  of  it  is  retained  for  months,  perhaps 
for  years,  unless  removed  by  similar  mechanical  violence 
applied  in  an  opposite  way.  This  condition  is  known  as  one 
of  subpermanent  magnetism. 

A  plate  of  iron  has  magnetism  induced  in  it  in  a  similar 
way,  the  magnetism  being  divided  into  regions  of  opposite 
polarity  by  a  neutral  plane  at  right  angles  to  the  direction  of 
the  earth's  total  force;  and  its  permanency  is  dependent  on 
the  character  of  the  iron  and  the  treatment  given  it. 

The  law  of  induction,  as  explained  for  rods  and  plates,  ex- 


MAGNETIC  INDUCTION  79 

tends  to  bodies  of  a  third  dimension,  whether  of  regular  or 
irregular  shape ;  the  line  connecting  the  induced  poles,  called 
the  magnetic  axis,  lying  in  the  direction  of  the  line  of  force, 
with  a  neutral  area  whose  plane  is  perpendicular  to  that  axis. 

63.  Magnetic  induction  in  an  iron  or  steel  ship. — Apply- 
ing this  law  of  induction  to  bodies  of  even  the  varied  and 
complex  form  of  an  iron  or  steel  ship,  it  is  easy  to  understand 
how  such  a  vessel  should  receive  magnetism  by  induction  and 
have  it  partially  fixed  in  the  course  of  construction  by  the 
processes  of  bending,  twisting,  hammering,  or  riveting  to 
which  the  various  parts  are  subjected. 

Since  the  facility  with  which  the  induction  takes  place  and 
the  ability  of  the  iron  to  retain  the  magnetism  induced  de- 
pend both  on  the  character  of  the  iron  and  the  treatment  it 
receives,  it  is  convenient  to  consider  separately  the  earth's 
effect  on  the  soft  and  hard  iron,  and  also  their  effects  on  the 
compass  needle. 

The  iron  in  which  only  temporary  magnetism  is  induced 
consists  of  the  kind  denominated  as  "  soft  iron/'  and  in  a  ship 
this  is  either  horizontal  or  vertical,  or  if  not  so,  it  may  be 
resolved  with  components  in  those  planes  so  that  the  earth's 
effect  on  soft  iron  will  be : 

(1)  Transient  magnetism  induced  in  horizontal  soft  iron, 
or  that  developed  in  the  horizontal  soft  iron  of  the  ship  by 
the  inductive  action  of  H,  the  horizontal  component  of  the 
earth's  total  force. 

It  is  transient  in  character  and  as  it  depends  for  its  force 
upon  H'j  which  varies  with  the  cosine  of  the  dip,  its  force  will 
be  zero  at  the  magnetic  poles  and  a  maximum  at  the  magnetic 
equator. 

(2)  Transient  magnetism  induced  in  vertical  soft  iron,  or 
that  developed  in  the  vertical  soft  iron  of  the  ship  by  the  in- 
ductive action  of  Z,  the  vertical  component  of  the  earth's 
total  force.     It  is  transient  in  character  and  as  it  depends  for 


80  NAVIGATION 

its  force  upon  Z,  which,  varies  with  the  sine  of  the  dip,  its 
force  will  be  zero  at  the  magnetic  equator  and  a  maximum  at 
the  magnetic  poles. 

Subpermanent  magnetism  induced  in  the  ship  while  build- 
ing.— The  remainder  of  the  ship's  iron,  consisting  of  that 
denominated  "hard  iron"  and  of  that  of  a  character  inter- 
mediate between  hard  and  soft  iron,  when  acted  upon  by  the 
earth's  inducing  forces  in  the  process  of  building,  assumes  the 
character  of  a  large  magnet,  more  or  less  permanent,  whose 
distribution  of  magnetism  depends  on  the  place  of  building 
and  the  azimuth  of  the  ship's  head  at  the  time.  Whilst  build- 
ing, the  ship's  polar  axis  and  neutral  plane  respectively  cor- 
respond, more  or  less,  to  the  direction  of  the  earth's  total  force 
and  a  plane  at  right  angles  to  it.  The  magnetism  thus  de- 
veloped is  known  as  subpermanent  magnetism,  as  it  is  not 
entirely  permanent;  suffering  a  diminution,  after  launching 
of  the  vessel  and  with  change  of  direction  from  that  in  which 
the  ship  was  built,  for  a  lapse  of  several  years  till  its  mag- 
netism settles  down  to  practically  a  permanent  condition. 
This  state  of  affairs  is  in  no  wise  due  to  induction  in  soft 
iron,  and  is  modified  if  the  vessel  is  launched  before  its  hull 
is  practically  completed. 

64.  Forces  acting  on  a  compass  needle  in  an  iron  or  steel 
ship. — It  is  evident  then  that  a  compass  needle,  besides  being 
acted  upon  by  the  earth's  horizontal  force  which  tends  to 
keep  the  needle  in  the  magnetic  meridian,  is  subject  to  three 
distinct  disturbing  influences  derived  from  the  ship  itself : 

(1)  Subpermanent  magnetism ; 

(2)  Transient  magnetism  due  to  vertical  induction  in  ver- 
tical soft  iron; 

(3)  Transient  magnetism  due  to  horizontal  induction  in 
horizontal  soft  iron ; 

and  that  the  resultant  of  these  three  forces,  when  not  acting 
in  the  plane  of  the  magnetic  meridian,  will  deflect  the  needle, 


SUBPERMANENT    MAGNETISM  81 

producing  the  total  deviation  for  the  particular  heading  of 
the  ship. 

65.  Effect  in  producing  deviation  of  each  of  the  ship's 
disturbing  forces  when  acting  on  the  compass  needle. 

(1)  Effect  of  subpermanent  magnetism. — We  have  seen 
that  the  location  of  the  poles  of  this  subpermanent  magnetism 
depends  upon  two  things:  1st,  the  magnetic  azimuth  of  the 
ship's  head  while  building;  2d,  the  direction  of  the  line  of 
force  at  the  place  of  building;  hence  the  North  (or  red)  pole 
will  be  in  that  part  of  the  ship  which  was  North  in  building 
and  the  South  (or' blue)  pole  will  be  in  that  part  which  was 
South  in  building.  The  repulsion  of  the  North  pole  of  the 
ship  simply  doubles  the  attraction  of  the  South  pole  for  the 
North  end  of  the  compass  needle,  therefore  it  may  be  laid 
down  as  a  general  rule  that,  under  the  influence  of  the  sub- 
permanent  magnetism,  the  North  end  of  the  compass  needle 
will  be  attracted  to  that  part  of  an  iron  or  steel  ship  which 
was  South  in  building;  hence  in  an  iron  ship  built  head 
North,  the  North  end  of  the  needle  will  be  attracted  toward 
the  stern.  Heading  N.  or  S.  there  will  be  no  deviation;  in  the 
former  case  the  directive  force  is  diminished,  in  the  latter 
case,  increased.  As  the  ship  swings  in  azimuth  from  these 
neutral  points  the  needle  is  deflected ; '  toward  the  East  for 
westerly  headings  with  a  maximum  of  deviation  about  West, 
toward  the  West  for  easterly  headings  with  a  maximum  about 
East. 

In  an  iron  ship  built  head  South,  the  North  end  of  the  com- 
pass needle  will  be  attracted  toward  the  stem  or  head  of  the 
ship,  and  results  just  the  reverse  of  the  above  will  be  ob- 
tained. 

In  an  iron  ship  built  head  East,  the  North  end  of  the 
compass  needle  will  be  attracted  to  the  starboard  side;  head- 
ing East  or  West  there  will  be  no  deviation,  in  the  former 
case  the  directive  force  is  diminished,  in  the  latter  case  it  is 


82 


NAVIGATION 


increased.  As  the  ship  swings  in  azimuth  from  these  neutral 
points,  the  compass  needle  will  be  deflected  toward  the  East 
for  northerly  headings  with  a  maximum  about  North,  toward 
the  West  for  southerly  headings  with  a  maximum  about  South. 


In  an  iron. ship  built  head  West,  the  North  end  of  the 
needle  will  be  attracted  toward  the  port  side,  and  results  the 
reverse  of  those  in  a  ship  built  head  East  will  be  obtained. 

The  accompanying  diagrams  will  illustrate  the  distribution 
of  magnetism  in  ships  built  head  N".,  E.,  S.,  and  W.  (mag- 
netic) at  a  place  where  the  magnetic  dip  is  68°  30',  and  the 


SEMICIRCULAR  DEVIATION 


83 


character  of  the  deviations  due  to  the  subpermanent  magnetism 
of  these  four  ships  is  illustrated  in  the  following  curves.  In 
reading  these  curves,  the  azimuths  are  taken  on  the  vertical 
line  and  the  deviations  on  the  ordinates  perpendicular  thereto, 
the  curve  being  to  the  right  of  the  vertical  line  when  the 
deviations  are  easterly,  to  the  left  when  westerly. 

Estimating  the  azimuth  of  the  ship's  head  from  the  neutral 
points,  these  curves  are  "  curves  of  sines " ;  they  show  the 
deviation  to  be  the  same  in  amount,  but  of  opposite  sign,  on 


points  differing  180°  in  azimuth,  and  the  neutral  points  to 
correspond  to  those  points  of  the  compass  on  which  the  ship's 
head  and  stern  were  in  building. 

Semicircular  deviation. — The  above  is  called  a  semi- 
circular deviation  because  it  is  easterly  in  one  semicircle  and 
westerly  in  the  other,  as  the  ship's  head  moves  around  a  com- 
plete circle  in  azimuth. 

If  the  compass  is  principally  disturbed  by  the  magnetic  in- 
fluences of  the  hull  of  the  ship,  the  neutral  points,  or  points 
of  no  deviation,  will  be  opposite  to  each  other  and  will  cor- 


84  NAVIGATION 

respond  to  those  points  of  the  compass  towards  which  the 
ship's  head  and  stern  were  directed  in  building;  and  the  de- 
viation, or  more  exactly  the  sine  of  the  deviation,  in  each  semi- 
circle, is  proportional  to  the  sine  of  the  azimuth  of  the  ship's 
head  measured  from  the  neutral  point,  the  azimuth  being  that 
shown  by  the  disturbed  compass. 

Since  the  force  due  to  subpermanent  magnetism  is  con- 
stant for  all  latitudes,  and  its  effect  in  producing  deviation 
is  inversely  as  H,  the  directive  force  of  the  earth,  the  semi- 
circular deviation  due  to  subpermanent  magnetism  varies  with 
change  of  latitude. 

66.  (2)  Effect  of  transient  magnetism  due  to  vertical  in- 
duction in  vertical  soft  iron. — The  vertical  component  of  the 
earth's  force,  Z,  induces  magnetism  in  the  vertical  soft  iron 
and  fittings  of  the  ship,  producing  a  resultant  pole  of  South 
polarity  towards  which  the  North  end  of  the  compass  needle 
is  attracted.  As  the  vertical  inducing  force  remains  the  same 
at  a  given  place,  the  magnetism  induced  by  it  does  not  vary 
as  the  ship  turns  in  azimuth;  therefore,  it  produces  a  semi- 
circular deviation  following  the  same  law  as  that  caused  by 
subpermanent  magnetism,  with  the  exception  that  the  effect 
produced  in  this  case,  being  directly  proportional  to  Z  and  in- 
versely proportional  to  H,  varies  as  the  tangent  of  the  mag- 
netic dip,  or  as  tan  6. 

The  semicircular  deviation  caused  by  induction  in  ver- 
tical soft  iron  is  the  kind  formerly  found  in  wooden  ships; 
the  neutral  points  being  North  and  South,  and  the  deviation 
easterly  in  the  Eastern  semicircle,  westerly  in  the  Western 
semicircle.  It  constitutes  the  smaller  part  of  the  semicir- 
cular deviation  of  iron  or  steel  ships. 

The  ship's  polar  force  and  the  starboard  angle. — As  the  two 
forces  we  have  just  considered,  those  due  to  subpermanent 
magnetism  and  transient  magnetism  induced  in  vertical  soft 
iron,  produce  the  same  kind  of  deviation,  it  is  convenient  to 


SEMICIRCULAR  DEVIATION 


85 


take  them  jointly  and  to  consider  the  North  point  of  the 
needle  as  acted  upon  by  their  resultant  force,  known  as  the 
ship's  polar  force;  its  horizontal  component  makes  with  the 
fore  and  aft  line  of  the  ship  an  angle,  which  measured  from 
ahead  around  to  the  right  (from  0°  to  360°)  is  known  as 
the  starboard  angle  and  designated  by  the  letter  a. 

The  semicircular  component  forces  and  the  coefficients  of 
semicircular  deviation. — Now  this  resultant  polar  force  of 
South  polarity,  attracting  the  North  end  of  the  compass 
needle  toward  a  certain  point  in  the  ship,  may  itself  be  re- 
solved into  two  component  forces,  one  acting  in  the  fore-and- 
aft  line  of  the  ship,  the  other  in 
the  athwartship  line  through  the 
compass.  Let  93  represent  the 
semicircular  force  acting  in  the 
fore-and-aft  line  and  &  the  semi- 
circular force  in  the  athwartship 
line;  the  former  is  marked  +  ^ 
acting  toward  the  ship's  head, 
( — )  if  toward  the  stern ;  the  lat- 
ter is  marked  +  ^  acting  to  star- 
board, ( — )  if  to  the  port  side. 
The  signs  of  these  forces  are  de- 
pendent  on  the  value  of  the  star- 
board angle  a  as  indicated  in 
Fig.  25. 

93  and  (£  are  known  as  the  exact  coefficients  of  the  semi- 
circular deviation;  but  +'  93  is  the  ship's  polar  force  to  head 
and  +  (£  is  the  ship's  polar  force  to  starboard,  both  expressed 
in  terms  of  the  "  mean  value  of  the  force  of  earth  and  ship 
to  magnetic  North"  as  a  unit.  This  fact  will  be  apparent 
from  a  study  of  section  II,  chapter  IV. 

Let  0  be  the  position  of  a  compass ;  OS,. ,  OS* ,  OSS ,  or 
OSi  be  a  ship's  polar  force  represented  in  Fig.  25  as  acting 


FIG.  25. 


86  NAVIGATION 

respectively  from  the  starboard  bow,  starboard  quarter,  port 
quarter,  or  port  bow,  and  in  the  horizontal  plane. 

For  the  force  08^  or  0£4 ,  23  =  Om,  for  the  force  OSa  or 
08S ,  3$  =  On;  the  sign  of  23  is  +  when  acting  to  head,  (— ) 
when  acting  to  stern  as  indicated  in  the  figure. 

For  the  forces  08^ ,  08 2 ,  08 3 ,  and  <9S4 ,  the  values  of  <£ 
are  represented  by  mS^,  nS2,  nSs ,  and  ra£4  respectively; 
the  sign  of  (£  is  +  when  acting  to  starboard,  ( — )  when  act- 
ing to  port  as  indicated  in  the  figure. 
For  the  force 

OS±,%is    +  ,  (£is    +  ,  and  a  is  <   90°, 
OS2,  $  is  (—),£  is    +  ,  and  a  is  >   90    and  <180°, 
OSS,  %  is  (— ),  (£  is  (— ),  and  a  is  >180    and  <270  , 
0&4,  23  is    +  ,  (£  is  (— ),  and  a  is  >270    and  <360  , 

all  values  of  a  being  measured  from  ahead  around  to  the  right, 
as  shown  by  circles  in  the  figure. 

It  is  convenient  to  find  the  angle  x  =  tan-1  -J-  and  then  to 

Jo 

take  a  =  x,  180°—  x,  180°  +  x,  or  360° --a,  according  as 
the  polar  force  acts  from  the  starboard  bow,  starboard  quar- 
ter, port  quarter,  or  port  bow  respectively. 

For  illustration,  consider  the  two  components  positive.  The 
forces  33  and  (£  exert  each  an  attraction  on  the  North  end  of 
the  compass  needle  similar  to  that  of  a  permanent  magnet, 
producing  semicircular  deviation  as  the  ship  swings  in  azi- 
muth. The  force  +  23  causes  no  deviation  on  the  heading 
North,  but  as  the  ship  swings  toward  the  East,  the  needle 
deviates  toward  the  East,  and  the  deviation  increases  with  the 
azimuth  by  constant  increments  till  the  ship  heads  about  East, 
when  the  force  is  at  right  angles  to  the  direction  of  the 
needle  and  produces  its  maximum  effect ;  then  as  the  swinging 
continues  the  deviation  diminishes  by  constant  decrements 
till,  when  the  ship  heads  South,  the  deviation  is  zero  and  the 
needle  is  again  in  the  magnetic  meridian.  If  the  swinging 


SEMICIRCULAR  DEVIATION 


87 


is  continued  through  the  western  semicircle,  the  effect  is  re- 
peated except  that  the  deviation  is  westerly  and  a  maximum 
in  amount  at  West. 

Letting  the  maximum  deviation  produced  be  represented  by 
B  (which  is  approximately  the  snr1  23),  the  deviation  on 
any  other  heading  due  to  23  will  be  a  fraction  of  B,  the  amount 
and  sign  depending  on  the  azimuth  of  the  ship's  head  per 


NW 


FIG.  28.      FIG.  29. 


compass.    B  is  known  as  the  approximate  coefficient  of  semi- 
circular deviation  due  to  the  force  93. 

If  as  in  Fig.  26,  the  azimuths  are  read  off  on  the  vertical 
line,  and  if,  at  the  points  representing  the  azimuths  of  the 
ship's  head  per  compass,  ordinates  are  erected  perpendicular 
to  the  vertical  line  and  representing  according  to  a  given  scale 
of  parts  the  corresponding  deviations,  the  curve  drawn 
through  the  extremities  of  the  ordinates,  showing  the  devia- 
tion to  be  zero  at  North  and  South  and  a  maximum  at  East 
and  West,  will  be  a  curve  of  sines ;  and  the  value  of  the  devia- 


88  NAVIGATION 

tion  due  to  the  fore-and-aft  semicircular  force  23  for  any 
heading  of  the  ship  z'  per  compass  will  be 

B  sin  z'. 

Had  the  force  23  attracted  the  North  end  of  the  needle  to 
the  stern,  or  if  the  sign  of  23  had  been  ( — ),  a  curve  similar 
to  Fig.  27  would  have  resulted,  the  deviations  being  westerly 
for  those  headings  on  which  the  force  +  23  produced  easterly 
deviations  and  easterly  for  those  on  which  +  %$  produced 
westerly  deviations. 

The  force  -(-  &  causes  a  maximum  easterly  deviation  when 
heading  North  which  diminishes  as  the  ship  swings  in  azi- 
muth to  the  eastward  till,  when  heading  East,  the  force  is  in 
the  magnetic  meridian  and  produces  no  deviation.  As  the 
swinging  is  continued,  the  North  end  of  the  needle  deviates 
to  West  and  the  amount  of  the  deviation  increases  by  con- 
stant increments  till,  when  the  ship  heads  South,  the  deviation 
is  again  a  maximum  in  amount  but  westerly  in  sign. 

If  the  swinging  is  continued,  this  effect  is  repeated,  the 
sign  of  the  deviation  being  opposite  to  that  of  the  corre- 
sponding point  180°  distant  in  azimuth. 

Letting  the  maximum  deviation  be  represented  by  C  (which 
is  appr®ximately  sin'1  (£),  the  deviation  on  any  other  heading 
due  to  +  (£  will  be  a  fraction  of  (7,  the  amount  and  sign 
depending  on  the  azimuth  of  the  ship's  head  per  compass. 
C  is  known  as  the  approximate  coefficient  of  semicircular  de- 
viation due  to  +  ®>  "the  athwartship  semicircular  force. 

As  seen  in  Fig.  28,  the  curve  is  one  of  cosines,  and  the 
value  of  the  deviation  on  any  heading  (zf)  per  compass  due 
to  the  force  +  ®  will  be 

C  cos  z', 

and  the  deviation  on  that  compass  heading  due  to  the  result- 
ant of  the  semicircular  forces  +  23  and  +  (£,  or  V^32  +  &2> 
will  be 

B  sin  z'  -\-  C  cos  z'. 


TRANSIENT  MAGNETISM  89 

67.  (3)  Effect  of  transient  magnetism  due  to  horizontal 
induction  in  horizontal  soft  iron. — Since  the  magnetism  in- 
duced in  horizontal  soft  iron  by  the  earth's  horizontal  force 
varies  as  the  cosine  of  the  inclination  of  the  iron  to  the  direc- 
tion of  the  earth's  horizontal  force,  it  is  evident  that  we  must 
now  deal  with  an  induced  force  which,  unlike  the  forces 
producing  semicircular  deviation,  is  a  variable  one. 

That  the  force  induced  may  be  a  function  of  the  azimuth  of 
the  ship's  head,  the  horizontal  soft  iron  of  the  ship  should  be 
considered  as  lying  either  in  the  fore-and-aft,  or  the  athwart- 
ship  direction  through  the  compass.  As  a  greater  part  of 
the  horizontal  soft  iron  is  so  situated,  and  as  soft  iron  at 
intermediate  angles  may  be  represented  by  components 
parallel  to  those  directions,  the  effect  of  all  the  soft  iron  in 
the  ship,  magnetized  by  the  earth's  horizontal  component  and 
acting  in  the  horizontal  plane  through  the  compass,  may  be 
replaced  by  the  effects  of  two  systems  of  horizontal  iron,  one 
placed  fore  and  aft  and  one  athwartships.  It  will  be  shown 
later  on  that  when  the  soft  iron  is  symmetrically  distributed 
on  each  side  of  the  fore-and-aft  line  of  the  ship,  and  the  ship 
is  on  an  even  keel,  that  the  forces  due  to  the  induced  mag- 
netism of  the  two  systems  may  be  replaced  by  those  of  a 
single  fore-and-aft  and  a  single  athwartship  rod;  the  force 
due  to  the  induced  magnetism  of  the  first  rod  will  act  in  the 
fore-and-aft  horizontal  line,  and  the  force  due  to  the  induced 
magnetism  of  the  second  rod  will  act  in  the  athwartship  hori- 
zontal line  through  the  compass. 

The  character  of  each  force,  whether  attracting  or  repell- 
ing the  North  end  of  the  compass  needle,  will  depend  on  the 
position  of  the  rod ;  if  entirely  forward  or  abaft,  to  starboard 
or  to  port,  of  the  compass,  one  end  of  the  rod  will  act ;  if 
continuously  extending  above  or  below  the  compass  needle, 
the  opposite  end  will  act.  If  a  rod  is  entirely  on  one  side  of 
the  compass,  a  similar  rod,  similarly  situated  but  distant  180°, 
will  simply  double  the  effect  of  the  first  rod. 


90  NAVIGATION 

Depending  on  the  location  of  the  compass  on  board,  the 
effect  of  the  symmetrical  horizontal  soft  iron  may  be  that  of 
one,  or  another,  of  the  arrangements  of  the  rods  as  shown  in 
Figs.  30,  31,  32,  and  33. 

Taking  the  first  rod  (1)  for  consideration  and  regarding 
the  magnetism  induced  in  the  rod  on  the  various  headings  as 
represented  in  Fig.  34,  we  see  that  with  the*  ship  heading 
North  no  deviation  is  produced  but  the  directive  force  is  in- 
creased; as  the  ship  swings  to  the  eastward,  the  North  point 
of  the  needle  deviates  to  the  East  (or  to  the  right) ;  at  NE., 
the  maximum  easterly  deviation  is  caused,  since  at  that  angle 
with  the  meridian,  the  induced  magnetic  force,  though  equal 
only  to  H  cos  45°,  has  a  greater  proportionate  effect  in  caus- 
ing deflection  of  the  needle.  As  the  ship  continues  to  swing 


— O-^-      -e1- 


2 


3 


FIG.  30.  FIG.  31.  FIG.  32.  FIG.  33. 

to  the  eastward,  the  rod  gradually  loses  its  magnetism,  and 
the  attraction  on  the  needle  diminishing  it  gradually  returns 
toward  the  meridian.  When  the  ship  heads  East,  the  rod  is 
at  right  angles  to  the  line  of  force  and  has  no  effect  on  the 
needle  which  is  again  in  the  meridian.  For  the  NE.  quadrant 
the  curve  of  deviations  is  as  represented  in  Fig.  35  from  North 
to  East. 

The  ship  continuing  to  swing,  the  end  of  rod  (1)  nearest 
the  compass,  having  North  magnetism  induced  in  it,  will 
repel  the  North  end  of  the  compass  needle,  and  the  curve  of 
deviations  traced  will  be  exactly  the  same  as  in  the  NE.  quad- 
rant except  that  it  will  be  on  the  opposite  side  of  the  vertical 
line.  When  the  ship  heads  South  there  is  no  deviation,  but  the 
directive  force  is  again  increased.  From  South  to  West  we 


QUADRANTAL    DEVIATION 


91 


will  have  the  same  curve  (the  deviations  being  the  counter- 
part in  amount  and  sign)  as  from  North  to  East,  at  West  the 
needle  fails  of  effect.  From  West  to  North  the  deviations 
and  the  curve  are  the  same  as  in  the  SE.  quadrant. 

duadrantal  deviation. — The  deviation  here  illustrated  is 
known  as  quadrantal  deviation,  and  is  so  called  because  it  is 
easterly  and  westerly  alternately  in  the  four  quadrants  as  the 
ship's  head  moves  around  a  complete  circle  in  azimuth.  Its 
zero  points  coinciding  very  nearly  with  the  cardinal  points, 


N.E. 


s  w. 


S.E. 


and  the  points  of  maximum  deviation  being  at  the  quadrantal 
points,  'each  characteristic  occurring  twice  in  a  semicircle, 
the  amount  of  the  deviation  (or  more  properly  the  sine  of  its 
amount)  is  proportional  to  the  sine  of  twice  the  azimuth  of 


92 


NAVIGATION 


the  ship's  head  measured  (as  will  be  seen  later)  from  a  line 
half  way  between  the  magnetic  North  and  the  compass  North. 
The  quadrantal  deviation  usually  found  on  board  iron  or  steel 
ships  is  of  the  type  represented  by  the  curve  of  Fig.  35,  easterly 
in  the  NE.  and  SW.  quadrants,  westerly  in  the  SE.  and  NW. 
quadrants,  or  the  type  usually  spoken  of  as  a  positive  quad- 
rantal  deviation. 

The  horizontal  component  of  the   earth's  total  magnetic 


North 


N.E. 


S.W. 


East 


S.E. 


South 


East 


S.E 


South 


West 


N.W. 


North 

FIG.  35. 


West 


N.W. 


North 

FIG.  36. 


North 


N.E. 


force  is  the  directive  force  acting  on  the  compass  needle,  the 
magnetism  induced  in  the  horizontal  iron  by  the  earth's  hori- 
zontal force  is  the  disturbing  force  acting  on  the  needle,  and 
the  ratio  between  the  two  is  constant;  therefore  the  quad- 


QUADRANTAL    DEVIATION  93 

rantal  deviation  may  be  expected  to  remain  unchanged  in  all 
magnetic  latitudes,  and  even  by  lapse  of  time,  so  long  as  the 
distribution  of  the  ship's  horizontal  soft  iron  remains  un- 
changed. 

Quadrantal  force  and  coefficient  of  quadrantal  deviation. — 
Let  2)  represent  the  quadrantal  force,  the  sign  of  the  force 
being  +  when  producing  a  positive  quadrantal  curve  as 
shown  by  Fig.  35;  and  letting  D  (which  is  approximately 
sin'1  3))  represent  the  maximum  deviation,  or.  that  on  the 
quadrantal  points,  the  deviation  on  any  other  heading  due  to 
the  force  +  3)  will  be  a  fraction  of  D  and  a  function  of  twice 
the  azimuth;  in  other  words  D  is  the  approximate  coefficient 
of  quadrantal  deviation  of  the  type  represented  by  the  curve 
(Fig.  35).  (See  Art.  79.) 

If  D-L  represents  the  coefficient  of  quadrantal  deviation  due 
to  the  induced  magnetism  of  rod  (1),  then  the  quadrantal 
deviation  on  any  heading  z'  per  compass  due  to  the  effect  of 
rod  (1)  will  be 

D!  sin  2z'. 

Taking  the  rod  (2)  under  consideration  and  regarding  the 
magnetism  induced  in  the  rod  on  the  various  headings  as  rep- 
resented in  Fig.  34,  it  is  seen  that  when  the  ship  heads  North 
no  effect  is  produced,  the  rod  being  at  right  angles  to  the 
line  of  force;  as  the  ship  swings  to  eastward,  polarities  are 
developed  as  shown  and  the  North  end  of  the  needle  is  re- 
pelled until,  when  the  ship  heads  NE.?  there  is  a  maximum 
westerly  deviation.  When  the  ship  heads  East,  the  rod  is  in  the 
meridian,  and  its  induced  magnetism  increases  the  directive 
force  on  the  needle,  but  produces  no  deviation.  It  is  thus 
seen  that  the  effect  of  the  magnetism  induced  in  a  rod  lying 
wholly  on  one  side  of  the  compass,  as  the  ship  swings  from 
North  to  East,  is,  in  the  case  of  an  athwartship  rod,  just  the 
reverse  of  that  in  a  fore-and-aft  rod  (1).  If  the  swinging 
is  continued  till  the  ship  again  heads  North,  the  deviations 
on  the  various  headings  may  be  represented  by  a  curve  simi- 


94  NAVIGATION 

lar  to  Fig.  36,  which  shows  for  any  azimuth  of  the  ship's 
head  a  deviation  of  the  same  kind,  but  of  the  opposite  sign,  to 
that  shown  by  the  curve  of  Fig.  35  as  in  the  case  of  rod  (1). 
When  the  ship  heads  West  the  directive  force  is  increased  as 
was  the  case  when  the  ship  headed  East. 

The  quadrantal  deviation  caused  by  rod  (2),  being  westerly 
in  the  N"E.  and  SW.  quadrants  and  easterly  in  the  SE.  and 
NW.  quadrants,  is  known  as  a  negative  quadrantal  deviation. 

In  this  case,  the  case  of  a  negative  quadrantal  deviation,  if 
( — )  D2  is  the  quadrantal  coefficient,  the  quadrantal  devia- 
tion on  any  heading  zf  per  compass  due  to  the  induced  mag- 
netism of  an  athwartship  rod  (2)  will  be 
—  D2  sin  2z'. 

The  combined  quadrantal  deviation  due  to  an  arrangement 
of  iron  similar  to  that  illustrated  in  Fig.  30  will  be 

(+!>!  +  (—  D2) )  sin  2zr  =  ±  D  sin  2z'. 
The  deviation  being  easterly  or  westerly  in  the  NE.  and  SW. 
quadrants  (and  hence  westerly  or  easterly  in  the  SE.  and  NW. 
quadrants),  according  as  the  effect  of  rod  (1)  is  greater  or 
less  than  the  effect  of  rod  (2).  In  cases  where  the  rod  (2) 
equals  rod  (1),  an  arrangement  of  iron  similar  to  Fig.  30 
will  increase  the  directive  force  without  causing  deviation. 

A  rod  (3),  Fig.  31  and  Fig.  33,  continuously  extending 
above  or  below  the  compass  in  a  transverse  plane,  will  have  an 
effect  directly  the  opposite  of  rod  (2),  diminishing  the  direc- 
tive force  at  East  and  West  and  producing  a  positive  quad- 
rantal deviation  as  represented  by  Fig.  35. 

If  Dz  be  the  coefficient  of  quadrantal  deviation  due  to  rod 
(3),  then  the  quadrantal  deviation  on  any  heading  z'  per 
compass  will  be 

D3  sin  2z', 

and  the  combined  quadrantal  deviation  due  to  an  arrangement 
of  iron  similar  to  Fig.  31  will  be 

(£>!  +  D3)  sin  2zf  =  D  sin  2z'. 


QUADRANTAL    DEVIATION"  95 

A  rod  (4),  Fig.  32  and  Fig.  33,  continuously  extending 
above  or  below  the  compass,  in  the  fore-and-aft  plane,  will 
exert  on  the  North  poin.t  of  the  compass  needle  an  effect  just 
the  opposite  to  that  exerted  by  rod  (1)  in  the  same  plane  but 
wholly  on  one  side  of  the  compass,  diminishing  the  directive 
force  at  North  and  South  and  producing  a  negative  quad- 
rantal  deviation  as  represented  by  Fig.  36.  If  — Z>4  be  the 
coefficient  of  the  quadrantal  deviation  due  to  rod  (4),  then  the 
quadrantal  deviation  on  any  heading  z'  per  compass  due  to 
induced  magnetism  of  rod  (4)  will  be 
(_)  D4  sin  2z', 

and  the  combined  quadrantal  deviation  due  to  an  arrangement 
of  iron  similar  to  Fig.  32  will  be 

(—  D2  +  (—  Z>4) )  sin  2z'  =  —  D  sin  2zf, 
and  that  due  to  an  arrangement  similar  to  Fig.  33  will  be 

(Z>3  +  (—  Z>  J )  sin  2z'  =  ±  D  sin  2z', 
the  sign  of  D  depending  on  which  is  the  greater,  the  mag- 
netism due  to  rod   (3)    or  that  due  to  rod    (4).     In  this 
arrangement  when  rod  (3)  equals  rod  (4),  the  directive  force 
will  be  diminished,  but  no  deviation  will  be  produced. 

Hence,  we  have  for  the  general  expression  representing 
quadrantal  deviation,  on  any  heading  z'  per  compass,  due  to 
horizontal  induction  in  horizontal  soft  iron  symmetrically  sit- 
uated, when  D  is  the  approximate  coefficient,  the  term 

D  sin  2z'. 

The  quadrantal  deviation  as  represented  by  D  is  usually 
caused  by  the  action  of  rods  (1)  and  (3),  Fig.  31,  or  excess 
of  effect  of  (3)  over  (4),  Fig.  33,  since  it  is  usually  positive 
and  the  directive  force  on  the  needle  is  diminished. 

68.  In  case  the  soft  iron  is  unsymmetrically  situated  in 
the  horizontal  plane  through  the  compass,  an  additional  force 
due  to  horizontal  induction  in  fore-and-aft  iron  may  act  on 
the  needle,  and  it  may  be  represented  as  that  of  a  fore-and-aft 


96 


NAVIGATION 


rod  to  starboard  or  port  of  the  compass,  rod  (5),  Fig.  37; 
and  an  additional  force  due  to  horizontal  induction  in 
athwartship  iron  may  also  act  on  the  needle,  this  force  being 
represented  as  that  of  an  athwartship  rod  forward  or  abaft 
the  compass,  rod  (6),  Fig.  37. 


s.w, 


S.E. 


Eegarding  the  magnetism  induced  in  rod  (5)  as  represented 
in  Fig.  37  for  the  various  headings  per  compass,  it  is  seen 
that  with  the  ship  heading  either  North  or  South,  the  North 
point  of  the  compass  needle  will  have  a  maximum  deflection 
to  eastward,  being  most  strongly  attracted  in  the  first  case  by 
the  induced  South  pole,  and  equally  repelled  in  the  second 
case  by  the  induced  North  pole  of  the  rod  (5). 


QUADRANTAL  DEVIATION  97 

When  the  ship  heads  either  East  or  West,  the  rod  (5)  has 
no  effect.  At  the  quadrantal  points  the  deviation  is  the  same 
fraction  of  the  maximum  and  also  easterly.  The  curve  rep- 
resenting the  deviations  is  similar  to  the  plain  curve  of 
Fig.  38. 

From  Fig.  37,  it  is  seen  that  when  the  ship  heads  North  or 
South  rod  (6)  has  no  effect.  When  the  ship  heads  East  or 


West,  the  North  point  of  the  needle  will  have  a  maximum 
deflection  to  the  westward,  being  most  strongly  repelled  in 
the  first  case  by  the  North  pole  of  (6)  and  equally  attracted 
in  the  second  case  by  the  South  pole  of  (6).  At  the  quad- 
rantal points  the  deviation  is  the  same  fraction  of  the  maxi- 
mum and  is  also  westerly.  The  curve  of  deviations  in  this 
case  is  similar  to  the  dotted  curve  of  Fig.  38. 

When  the  two  rods   (5)   and   (6),  Fig.  37,  act  together, 
rod  (5)  causes  a  maximum  easterly  deviation  when  rod  (6) 


98  NAVIGATION 

has  no  effect,  and  rod  (6)  causes  a  maximum  westerly  devia- 
tion when  (5)  has  no  effect;  at  the  quadrantal  points  the 
easterly  deviation  caused  By  (5)  neutralizes  the  westerly  de- 
viation caused  by  (6) ;  the  resultant  curve  of  deviation,  as 
represented  in  Fig.  39,  shows  maxima  when  the  ship  heads 
on  the  cardinal  points  and  minima  when  the  ship  heads  on 
the  quadrantal  points. 

This  deviation,  due  to  horizontal  induction  in  soft  iron 
unsymmetrically  distributed  about  the  compass,  is  of  the 
quadrantal  type  and  is  in  general  very  small. 

Let  (£  represent  the  force  producing  this  particular  kind 
of  deviation,  -+-  when  it  causes  an  easterly  deviation  between 
North  and  NE.  as  shown  in  Fig.  39 ;  let  E  (which  is  approxi- 
mately sin'1  (?)  be  the  maximum  deviation,  or  that  on  the 
cardinal  points,  then  the  deviation  on  any  other  heading  z' 
per  compass  due  to  the  force  +  (£  will  be  a  fraction  of  E  and 
will  be  found  from  the  expression 

E  cos  2z'. 

E  is  known  as  the  approximate  coefficient  of  the  quadrantal 
deviation  due  to  horizontal  induction  in  horizontal  soft  iron 
unsymmetrically  distributed  about  the  compass.  ( See  Art.  80. ) 

If  the  rods  (5)  and  (6)  are  in  the  port  bow  (Fig.  40.),  the 
deviation  due  to  their  effect  on  any  azimuth  will  be  the  .same 
in  amount  but  of  the  opposite  sign  to  that  produced  in  the 
case  when  they  were  in  the  starboard  bow,  the  coefficient  will 
be  ( — )  Ej  and  the  deviation  on  any  heading  zf  per  compass 
will  be 

— E  cos  2z'. 

If  the  rods  are  both  in  one  quarter,  the  effect  will  be  the 
same  as  if  they  were  in  the  opposite  bow;  the  effect  of  having 
one  of  the  rods  in  the  bow  and  one  in  the  opposite  quarter  is 
the  same  as  if  both  rods  were  in  that  bow,  or  both  in  that 
quarter. 

So  the  general  expression  for  the'  quadrantal  deviation  on 


CONSTANT  DEVIATION 


99 


any  heading  z    per  compass  due  to  horizontal  induction  in 
soft  iron  unsymmetrically  distributed  about  the  compass  is 

E  cos  2sf. 

69.  Constant  deviation. — When  the  soft  iron  is  not  sym- 
metrically distributed  on  each  side  of  the  fore-and-aft  line 
through  the  compass,  or  the  compass  is  not  in  the  midship 
line,  a  constant  term  may  be  noted  in  the  deviation. 

It  is  usually  very  small  and  is  called  constant  because  it  is 
the  same  in  amount  and  direction  on  all  headings ;  it  is 
marked  East  or  (+)  when  the  easterly  deviation  is  in  excess, 
West  or  ( — )  when  the  westerly  deviation  is  in  excess.  If 
due  to  the  unsymmetrical  soft  iron  of  the  ship,  it  is  known 
as  a  real  constant  deviation  which  will  not  vary  with  change 
of  latitude,  as  the  ratio  of  the  force  induced  in  the  iron  and 
the  directive  force  on  the  compass  needle  is  a  constant.  The 


FIG.  40. 


FIG.  41. 


FIG.  42. 


force  producing  constant  deviation  is  represented  by  91,  the 
approximate  coefficient  or  deviation  itself  by  A.  (See  Art.  80.) 
If  rods  (5)  and  (6)  are  situated  as  in  Fig.  41,  (5)  will 
produce  the  plain  curve  of  Fig.  38  when  the  ship  is  swung, 
and  (6)  will  produce  a  curve  similar  to  the  dotted  curve  of 
Fig.  38  except  of  the  opposite  sign;  that  is  to  say  (5)  will 
cause  a  maximum  of  deviation  on  North  and  South,  a  mini- 


100  NAVIGATION 

mum  on  East  and  West,  all  being  easterly;  (6)  will  cause  a 
minimum  of  deviation  on  North  and  South,  and  a  maximum 
on  East  and  West,  also  easterly.  In  other  words,  as  the  effect 
of  one  rod  increases,  that  of  the  other  decreases,  their  result- 
ant effect  as  the  ship  swings  in  azimuth  being  a  constant 
easterly  deviation. 

If,  however,  rods  (5)  and  (6)  are  situated  as  in  Fig.  42, 
their  resultant  effect  as  the  ship  swings  in  azimuth  will  be  a 
constant  westerly  deviation.  A  real  value  of  A  is  rare  on 
board  ships  with  well-located  compasses;  more  often  there  is 
an  apparent  value  due  to  a  badly  placed  lubber's  line,  index 
or  other  instrumental  error,  or  an  error  in  the  assumed  direc- 
tion of  the  magnetic  North.  Other  errors  of  a  like  nature 
may  exist,  but,  whatever  the  causes,  they  may  all  be  repre- 
sented by  the  approximate  coefficient  of  constant  deviation  A. 

70.  If  8  represents  the  sum  total  of  the  deviations  due  to 
the  various  forces  considered  for  the  heading  z'  per  compass 
we  shall  have 

8  =  A  +  B  sin  z'  +  C  cos  z'  +  D  sin  2z'  +  E  cos  2z',     (16) 

A,  B,  C,  D,  and  E  being  known  as  the  approximate  coefficients 
of  the  deviation  they  respectively  represent.  (See  Art.  77.) 

Though  the  curve  of  total  deviations,  as  shown  on  Napier's 
diagram  (Art.  60),  is  irregular  and  unsymmetrical,  the  curves 
due  to  the  separate  forces  considered  are  in  themselves  per- 
fectly symmetrical,  the  irregularity  arising  from  the  super- 
position of  all  of  them,  the  combined  curve  representing  the 
resultant  effect  of  all  the  forces. 

71.  Determination  of  coefficients  by  inspection. — The  co- 
efficients A,  B,  G,  D,  and  E  may  be  found  from  the  .deviations 
observed  with  the  ship's  head  on  the  24  equidistant  15°  com- 
pass rhumbs;  also  on  any  12  or  8  of  them  if  equidistant,  as  will 
be  indicated  in  Art.  89,  the  process  there  followed  being 
known  as  the  "  analysis  of  deviations  " ;  but  A,  B,  C,  and  E 


APPROXIMATE  COEFFICIENTS  101 

may  be  approximately  determined  by  inspection,  from  the  de- 
viations on  the  four  cardinal  points,  and  D  may  be  also 
approximately  obtained  from  the  deviations  on  the  four  quad- 
rantal  points. 

Using  formula  (16),  Art.  70,  and  paying  particular  atten- 
tion to  the  signs  of  functions  of  the  azimuths,  we  have  equa- 
tions expressing  the  deviations  on  the  eight  principal  points 
as  follows,  letting  83  be  the  sine  or  cosine  of  45° : 

+  0  +  E 

+  C83  +  D 

—  C83  —  D 

-C  +E 

-C83  +  D 

—  E 
+  C83-D 

and  it  is  apparent  from  these  equations  that,  using  the  word 
mean  in  its  algebraic  sense, 

A  is  the  mean  of  the  deviations  on  the  four  cardinal  points, 
or  any  four  or  more  equidistant  compass  headings. 

B  is  approximately  the  deviation  at  East,  or  the  deviation 
at  West  with  the  sign  changed ;  but  more  accurately  the  mean 
of  these  two  values. 

C  is  approximately  the  deviation  at  North,  or  the  deviation 
at  South  with  the  sign  changed;  but  more  accurately  the  mean 
of  these  two  values. 

D  is  approximately  the  mean  of  the  deviations  at  NE.  and 
SW.,  or  the  mean  of  the  deviations  at  SE.  and  NW.  with  the 
sign  changed;  but  more  accurately  the  mean  of  both  these 
means. 

E  is  the  mean  of  the  deviations  on  the  four  cardinal  points 
of  the  compass  after  the  signs  of  the  deviations  on  East  and 
West  have  been  reversed. 


On 

North 

So 

=  A 

NE. 

S3 

=  A 

+  BS, 

East 

S6 

—  A 

+  B 

SE. 

So 

=  A 

+    B8; 

South 

812 

=  A 

SW. 

818 

=  A 

—  B8. 

West 

Sis 

—  A 

—  B 

NW. 

821 

—  A 

—  B8. 

102  NAVIGATION 

Ex.  7. — Given  the  following  deviation  table,  find  by  inspec- 
tion the  nearest  approximation  to  the  values  of  A,  B,  C,  D, 
and  E. 


North    3°  12'  E 
NE       14    24  E 
East    12    00  E 
SE         5    24   E 

South    1°  36'  E 
SW        7    00   W 
West   14    24   W 
NW      12    12   W 

To  find  A. 

To 

find  B. 

To  find  C. 

At  North    3°  12'  E 

At  East 

12°  00'  E 

At  North 

3°  12'  E 

East     12    00   E 

West 

(—  )  14    24   W 

South 

(—  )  1    36   E 

South    1    36   E 

West    14   24  W 

2)26°  24'  E 

2)  1°  36'  E 

Algebraic  Sum    2°  24'  E  B  =  13°  12'  E                                   C  =  0°  48'  E 
A=    2°  24'  E  =  QO  36>  E 

To  find  D.  To  find  E. 

At  NE  14°  24'  E  At  SE      5°  24'  E                   At  North            3°  12'  E 

SW    7    00   W  NW  12    12   W                        East     (— )  12    00   E 

South  1    36   E 

2)7°  24'  E  2)6°  48'  W                        West    (— )  14    24   W 

1st  value  of  D  =  3°  42'  E  2d  value  D  3°  24'  W 
2d  value  of  D(— )  3    24  W 


2)7°  06'  E 
D  =  3°  33'  E 

72.  Heeling  error. — So  far  only  those  forces  acting  on  the 
North  point  of  the  compass  needle  to  produce  deviation  when 
the  ship  is  upright  have  been  considered,  and  it  is  necessary 
to  consider  other  forces  when  an  iron  ship  is  inclined  from 
the  vertical. 

There  are  certain  forces  which,  acting  only  vertically  and 
producing  no  deviation  in  the  former  case,  will  have  in  the 
latter  case  lateral  components  tending  to  draw  the  North  point 
of  the  needle  to  one  side  or  the  other.  Such  are  the  vertical 
component  of  the  ship's  subpermanent  magnetism  and  the 
vertical  component  of  the  magnetism  induced  in  vertical  soft 
iron.  If  they  act  vertically  downward  when  the  ship  is  on  an 
even  keel,  the  North  end  of  the  needle  will  go  to  windward 
when  the  ship  heels,  otherwise  to  leeward. 

In  addition,  the  horizontal  deck  beams  and  all  other  horizon- 
tal transverse  iron  become  more  or  less  magnetized  by  the 
earth's  vertical  inducing  force,  and  the  South  polarity  of  their 


MEAN  DIRECTIVE  FORCE  103 

upper  or  weather  ends  will  attract  the  North  end  of  the 
needle  to  windward. 

The  resultant  effect  of  these  forces,  when  the  ship  is  in- 
clined, is  known  as  the  heeling  error,  the  direction  of  which,  de- 
pending on  circumstances,  may  be  to  windward  or  to  leeward. 

The  heeling  error  will  vary  with  change  of  latitude  because 
that  part  due  to  the  vertical  component  of  the  subpermanent 

magnetism  varies  as  -jf,  and  the  parts  due  to  magnetism  in- 
duced by  the  earth's  vertical  force  vary  as  tan  0. 

This  error  is  a  maximum  on  northerly  or  southerly  courses, 
a  minimum  on  easterly  or  westerly  courses,  and,  for  inter- 
mediate headings,  varies  practically  as  the  cosine  of  the  azi- 
muth of  the  ship's  head. 

The  causes  and  effects  of  heeling  error  may  be  better  under- 
stood after  a  careful  study  of  the  next  chapter. 

73.  Mean  directive  force. — The  ship's  forces  acting  on  the 
compass  have  been  considered  primarily  as  causing  deviation, 
but  they  have  an  additional  effect,  increasing  or  diminishing 
the  earth's  directive  force  on  the  various  headings  as  the  ship 
swings  in  azimuth.  This  can  be  easily  seen  by  resolving  any 
force  so  that  one  component  will  be  in  the  direction  of  the 
undisturbed  needle  and  one  at  right  angles  to  it  in  the  hori- 
zontal plane. 

Whilst  the  latter  component  acts  to  produce  deviation,  the 
former  increases  or  diminishes  the  directive  force  according 
as  it  draws  the  North  point  of  the  needle  to  the  northward  or 
southward. 

The  force  of  earth  and  ship  to  magnetic  North  will  vary 
with  the  azimuth  of  the  ship's  head,  and  its  mean  value  for 
equidistant  headings  will  be  the  mean  directive  force  acting 
on  the  needle  which,  experience  shows,  is  less  than  unity  (H 
being  unity)  in  nearly  all  iron  or  steel  ships. 

Other  conditions  being  the  same,  the  best  location  for  a 
compass  is  that  position  where  it  will  have  the  greatest  mean 
directive  force.  (See  Art.  76.) 


CHAPTER  IV. 

MATHEMATICAL    THEOKY    OF    THE    DEVIATIONS    OF 
THE  COMPASS. 

SECTION  I. 

74.  Mathematical  theory  of  the  deviations  of  the  compass. 

— By  considering  all  the  iron  of  a  ship  as  magnetically  either 
hard  or  soft,  it  has  been  shown  that  on  board  an  iron  or  steel 
ship,  a  freely  suspended  needle  is  acted  upon  by  (1)  the 
earth's  total  force;  (2)  the  force  due  to  the  subpermanent 
magnetism  induced  in  the  ship  in  building;  (3)  those  forces 
due  to  the  transient  magnetism  induced  in  soft  iron  by  the 
earth's  force. 

At  a  given  place  the  force  (1)  is  a  constant  force,  though  it 
does  not  draw  the  North  point  of  the  needle  towards  the  same 
point  in  the  ship  on  all  azimuths  of  the  ship's  head.  The 
force  under  (2)  is  a  constant  force  and  attracts  the  North 
point  of  the  needle  towards  the  same  point  in  the  ship  for  all 
azimuths.  The  forces  under  (3)  are  constant  or  variable,  de- 
pending on  whether  the  inducing  force  is  the  vertical  or  hori- 
zontal component  -of  the  earth's  force. 

In  investigating  mathematically  the  theory  of  compass  de- 
viations, it  is  necessary  first  to  find  the  components  of  the 
various  forces  acting  on  the  North  point  of  the  needle  in 
certain  definite  directions  through  the  compass,  and  the 
resultant  of  the  components  in  each  direction. 

Let  these  directions  be  the  fore-and-aft  horizontal  line, 
the  transverse  horizontal  line,  and  the  vertical  line  through 
the  point  of  suspension  of  the  needle,  the  length  of  which  is 


THEORY  OF  DEVIATIONS  OF  COMPASS 


105 


regarded  as  infinitely  small  when  compared  with  the  distance 
of  the  nearest  iron,  or,  what  amounts  to  practically  the  same 
thing,  the  North  point  of  the  needle  is  considered  the  origin 
of  coordinates. 

Examining  first  the  earth's  force,  let  0,  Fig.  43,  represent 
the  North  point  of  a  magnetic  needle  on  board  an  absolutely 


FIG.  43. 

non-magnetic  ship ;  let  it  be  the  origin  of  a  system  of  rectan- 
gular coordinates  of  which  OX  the  horizontal  fore-and-aft 
line,  OY  the  horizontal  transverse  line,  and  OZ  the  vertical 
line  through  the  point  0  are  the  axes  respectively  denominated 
X,  Y,  and  Z ;  the  directions  to  head,  to  starboard,  and  verti- 
cally downward  being  regarded  as  positive.  The  South  end 
of  the  needle  is  not  considered,  since  an  attraction  or  repulsion 
of  the  North  end  would  be  a  repulsion  or  attraction  of  the 
south  end,  and  the  effect  on  the  South  end  simply  doubles  that 
on  the  North  end  without  changing  the  direction  of  the  needle. 


106  NAVIGATION — COMPASS  DEVIATION 

Therefore,  we  may  confine  the  investigation  to  the  action  on 
the  North  point  of  the  needle  of  the  earth's  total  force  which, 
for  purposes  of  illustration,  may  be  represented  in  intensity 
and  direction  by  OT  in  the  figure.  The  vertical  plane  of  OT 
is  that  of  the  magnetic  meridian  OM',  the  angle  M'OT  is  the 
magnetic  dip,  OM  represents  in  intensity  and  direction  the 
earth's  horizontal  inducing  force,  and  OC  the  intensity  of  the 
earth's  vertical  force  through  the  origin  of  coordinates  (see 
equation  15,  Article  45). 

If  the  needle  is  a  freely  suspended  needle,  it  will  point  in 
the  direction  OT;  if  a  compass  needle,  it  will  point  in  the 
direction  OM;  in  other  words,  OM  is  the  directive  force  acting 
on  the  compass  needle,  and  is  in  the  horizontal  plane  of  XY. 

Since  a  force  represented  by  a  vector  or  line  of  given  length 
and  direction,  such  as  OM,  can  be  resolved  into  two  com- 
ponents at  right  angles  to  one  another,  all  being  in  the  same 
plane,  then  the  force  OM,  or  Hf  can  be  resolved  into  the  forces 
OA  and  OB. 

Therefore,  letting  M'OX  be  the  magnetic  azimuth  of  the 
ship's  head,  measured  positively  around  to  the  right  from  the 
magnetic  meridian,  then,  if  in  Fig.  43,  £'s  0AM,  BOA,  and 
OM T  are  right  angles,  we  shall  have  the  components  of  the 
earth's  total  force  in  the  direction  of  the  three  axes  as  follows : 

OA  =  H  cos  z,  force  of  the  earth's  magnetism  drawing 
the  North  point  of  the  needle  at  0  towards  the  ship's 
head; 

OB  =  H  cos  (90°  +  z)  =  —  H  sin  z,  the  force  draw- 
ing it  to  starboard ; 

OC  =  Z,  the  force  drawing  it  vertically  downward. 

Let  the  component  OA  in  the  axis  of  X  be  called  X,  the 
component  OB  in  the  axis  of  Y  be  called  Yt  the.  component 
OC  in  the  axis  of  Z  be  called  Z 


THEORY  OF  DEVIATIONS  OF  COMPASS  107 

Now,  if  the  earth's  total  force  be  disregarded,  and  forces 
equivalent  to  X,  Y,  and  Z  be  supposed  to  act  in  their  proper 
axes  on  the  North  point  of  a  freely  suspended  magnetic  needle 
at  0,  it  will  take  the  direction  of  OT ';  a  compass  needle  under 
the  same  forces  would  take  the  direction  OM. 

If  instead  of  being  mounted  on  board  an  absolutely  non- 
magnetic ship,  the  magnetized  needle  is  mounted  on  board  an 
iron  or  steel  ship  in  the  same  geographical  locality,  the  forces 
due  to  the  ship's  magnetism  will  act  on  the  North  point  of  the 
needle  in  addition  to  the  earth's  force;  a  dipping  needle  will 
take  the  resultant  direction  of  all  the  forces,  and  a  compass 
needle  the  direction  OM".  The  angle  M'OM"  is  the  deviation 
due  to  the  ship's  magnetic  forces  for  the  particular  heading 
per  compass  z' ' ,  and  is  marked  East  or  -f-  when  OM."  is  to  the 
right  of  OM',  that  is,  when  the  North  point  of  the  compass  is 
drawn  to  the  Eastward ;  otherwise,  West  or  ( — ) . 

As  the  force  due  to  the  subpermanent  magnetism  of  the 
hard  iron  of  the  ship  alters  neither  its  intensity  nor  its  direc- 
tion as  the  ship  swings  in  azimuth,  the  components  of  this 
force  in  the  three  axes  will  be  constant.  They  are  represented 

by 

P  when  drawing  the  North  point  of  the  needle  towards 
the  ship's  head, 

Q  when  drawing  the  North  point  of  the  needle  to  star- 
board, 

R  when  drawing  the  North  point  of  the  needle  vertically 

downward. 
Expressed  in  terms  of  the  earth's  horizontal  force,  we  have 

P    O    7? 

abstract  quantities  -^-,  ¥f>jj- 

The  greater  portion  of  the  soft  iron  on  board  ship  lies  in 
one,  or  another,  of  the  three  axes  considered,  and  since  the 
effect  of  soft  iron  ^ng  in  intermediate  directions  may  be 
represented  by  other  iron  parallel  to  the  three  axes,  the  inves- 


108  NAVIGATION — COMPASS  DEVIATION 

tigation  of  the  effect  of  soft  iron  may  be  confined  to  the  effects 
of  iron  parallel  to  those  axes,  and  may  properly  begin  with  a 
consideration  of  the  fore  and  aft  system.  It  is  plain  that  the 
northern  portions  of  this  iron  will  always  have  induced  in 
them  North  polarity  and  the  southern  portions  South  polarity 
as  the  ship  swings  through  a  complete  circle  in  azimuth,,  that 
the  intensity  of  the  force  due  to  the  induced  magnetism  will 
vary  with  the  azimuth  of  the  ship's  head,  and  that  the  polari- 
ties will  be  reversed  when  the  ship's  head  passes  through  an 
azimuth  of  90°  or  270°.  If  the  ship  heads  North  magnetic, 
the  force  induced  in  the  fore  and  aft  soft  iron  will  be  a  maxi- 
mum and  will  bear  a  certain  specific  ratio  to  the  earth's  hori- 
zontal force.  If  that  ratio  be  19  then  the  force  induced  will 
be  IH.  With  the  ship  on  any  magnetic  azimuth'  z,  the  in- 
duced force  will  be  IH  cos  z.  As  X  =  H  cos  z,  the  force  on 
the  heading  z  will  be  the  same  fraction  of  X  that  the  possible 
maximum  is  of  H,  and  as  in  this  case  it  is  desirable  to  express 
a  force  with  a  suggestion  of  the  axis  in  which  the  iron  may  be, 
it  is  convenient  to  represent  the  force  induced  in  the  fore- 
and-aft  iron  as  IX,  or  as  >a  fraction  of  the  fore-and-aft  com- 
ponent of  the  inducing  force;  furthermore,  the  variations  in 
the  force  induced  and  the  reversal  of  polarities  referred  to 
will  occur  in  the  same  order  and  at  the  proper  time  even 
though  X  is  taken  as  the  inducing  force  in  the  axis  of  X. 

Therefore,  it  is  mathematically  correct  to  assume  that  the 
soft  iron,  lying  in  the  direction  of  any  one  of  the  axes,  has 
magnetic  force  induced  in  it  by  the  earth's  resolved  com- 
ponent in  the  same  axis  and  by  that  component  only. 

The  fore-and-aft  component  X  will  induce  magnetism  in 
the  fore-and-aft  iron  which  may  practically  be  considered  as  a 
fore-and-aft  system  of  parallel  magnets  attracting  the  North 
point  of  the  needle  with  a  force  IX  towards  a  point  or  pole  in 
the  system,  I  being  a  constant  dependent  only  on  the  soft  iron 
in  the  ship. 


THEORY  or  DEVIATIONS  OF  COMPASS  109 

This  force  will  act  in  the  fore-and-aft  vertical  plane  through 
the  compass  only  when  the  iron  is  symmetrically  situated  with 
reference  to  that  plane,  and  in  the  fore-and-aft  horizontal 
line  only  when  the  iron  is  symmetrically  situated  with  refer- 
ence to  the  horizontal  plane  through  the  compass.  Such  is  not 
the  usual  case,  and  this  force,  instead  of  acting  towards  the 
ship's  head,  acts  in  some  other  direction. 
The  components  of  IX  will  be 

aX  to  the  ship's  head, 

dX  to  starboard, 

gX  vertically  downward. 

y ,  y,  and  I  are  the  direction  cosines  of  IX;  a,  d,  and  g  are 

not  forces  but  are  constant  ratios,  and  do  not  change  with 
azimuth  or  geographical  position.  These  values  depend  only 
on  the  amount,  arrangement,  and  capacity  for  induction  of 
the  soft  iron  of  the  ship. 

Thus  a  is  the  ratio  between  the  component  X  and  the  com- 
ponent in  the  same  axis  of  the  force  induced  in  fore-and-aft 
soft  iron;  d  is  the  ratio  between  the  component  X  and  the 
component  to  starboard  of  the  force  induced  in  fore-and-aft 
soft  iron  and  is  zero  when  that  iron  is  symmetrically  situated 
with  reference  to  the  fore-and-aft  vertical  section  through  the 
compass;  g  is  the  ratio  between  the  component  X  and  the 
component  downward  of  the  force  induced  in  fore-and-aft  soft 
iron. 

In  the  same  way  it  may  be  shown  that  the  magnetism  of 
the  transverse  iron  is  induced  in  it  by  Y  and  only  by  F,  and 
that  the  force  so  induced  is  a  specific  fraction  of  Y  with  com- 
ponents 

bY  to  the  ship's  head, 

eY  to  starboard, 

hY  vertically  downward. 

Also,  that  the  earth's  vertical  force  Z,  and  that  only,  will 


110  NAVIGATION — COMPASS  DEVIATION 

induce  a  force  in  the  vertical  soft  iron  which  will  be  a  specific 
fraction  of  Z ,  and  the  components  of  this  force  in  the  three 
axes  will  be 

cZ  to  ship's  head, 

•fZ  to  starboard, 

TcZ  vertically  downward. 

I,  e,  li,  c,  f,  and  Tct  like  a,  d,  and  g,  are  abstract  quantities 
or  constant  ratios  that  do  not  vary  with  change  of  azimuth  or 
geographical  position,  and  are  similarly  denned;  ~b,  f,  and  h 
become  zero  when  the  transverse  and  vertical  soft  iron  is  sym- 
metrically situated  with  reference  to  the  fore-and-aft  vertical 
plane  through  the  compass.  If  the  transverse  iron  is  so  situ- 
ated, and  extends  across  the  ship,  there  will  be  induced  a  pole 
of  North  polarity  on  one  side  of  the  compass  and  a  pole  of 
South  polarity  on  the  opposite  side  at  an  equal  distance  from 
the  compass  and  the  vertical  fore-and-aft  section  through  it. 
One  pole  will  repel  and  the  other  will  attract  the  North  point 
of  the  needle  with  an  equal  force,  the  components  of  which  in 
the  fore-and-aft  line  will  be  equal  and  of  opposite  sign;  in 
other  words,  b  will  be  zero. 

If  the  transverse  iron  does  not  extend  across  the  ship,  but  is 
broken,  the  nearer  pole  on  one  side  will  be  of  one  polarity  and 
the  nearer  pole  on  the  other  side  will  be  of  the  opposite  polar- 
ity, and  if  the  iron  is  symmetrical  with  reference  to  the  fore- 
and-aft  section  through  the  compass,  &  will  reduce  to  zero  for 
the  reasons  above  given. 

In  like  manner  it  may  be  shown  that  under  the  same  cir- 
cumstances the  components  in  the  vertical  direction  will  have 
equal  values  with  contrary  signs;  in  other  words,  h  will  re- 
duce to  zero. 

It  is  evident  that  /  will  be  zero  when  the  vertical  iron  is 
symmetrically  distributed  on  each  side  of  the  vertical  fore-and- 
aft  plane  through  the  compass,  since  the  pole  of  the  system 
will  lie  in  that  plane. 


THEORY  OF  DEVIATIONS  OF  COMPASS  111 

Fundamental  Equations. — It  follows  then  that  a  compass 
needle  on  board  an  iron  or  steel  ship  is  acted  on  by  the  follow- 
ing forces  whose  components  are  given  for  the  axes  of  X,  Y, 
and  Z,  in  order : 

(1)  The  earth's  magnetic  force  whose  components  in  the 
three  axes  are 

X,  Y,  and  Z. 

(2)  The  resultant  pole  of  the  ship's  subpermanent  mag- 
netism whose  components  are 

P,  Q,  and  R. 

(3)  The  magnetic  force  due  to  transient  induced  magnetism 
in  fore-and-aft  soft  iron  whose  components  are 

aX,  dX,  and  gX. 

(4)  The  magnetic  force  due  to  transient  induced  magnetism 
in  transverse  soft  iron,  whose  components  are 

IT,  eY,  and  hY. 

(5)  The  magnetic   force  due  to  transient  induced  mag- 
netism in  vertical  soft  iron  whose  components  are 

cZ,  fZ,  and  TeZ. 

Therefore,  if  Xf,  Y',  and  Z'  represent  respectively  the  com- 
bined forces  due  to  the  magnetism  of  earth  and  ship  in  the 
axes  of  X,  Y ,  and  Z  acting  on  the  North  point  of  the  needle, 
then 

X'  =  X  +  aX  +  IY  +  cZ  +  P.  (17) 

Y'  =  Y  +  dX  +  eY  +  fZ  +  Q.  (18) 

Z'  =  Z  +  gX  +  hY+JcZ  +  R.  (19) 

These  equations,  first  used  by  M.  Poisson  and  now  known 

by  his  name,  form  the  groundwork  of  all  equations  used  in  the 

mathematical  theory  of  the  deviations  of  the  compass. 

From  the  above  equations  it  is  plain  that  for  the  subper-' 
manent  magnetism  of  the  ship  one  or  more  permanent  mag- 
nets, P,  Q,  and  R,  may  be  substituted,  and  for  the  soft  iron 


112  NAVIGATION — COMPASS  DEVIATION 

of  the  ship,  whatever  may  be  its  amount  or  direction,  nine 
soft  iron  rods,  a,  ~b,  c,  d,  e,  f,  g,  h,  and  lc,  illustrated  in  Plate  III, 
may  be  substituted ;  the  magnetism  induced  in  the  rods  a,  ~b, 
and  c  will  produce  a  force  in  the  axis  of  X,  +  if  to  head, 
( — )  if  toward  the  stern ;  the  magnetism  induced  in  d,  e,  and  / 
will  produce  a  force  in  the  axis  of  Y,  +  if  to  starboard,  ( — ) 
if  to  port;  and  the  magnetism  induced  in  g,  h,  and  Tc  will 
produce  a  force  in  the  axis  of  Zt  +  if  vertically  downward, 
( — )  if  upward. 

Parameters. — The  quantities  a,  6,  c,  d,  e,  f,  g,  hf  ~k,  P,  Q, 
and  E  are  called  parameters  and  are  constant;  the  first  nine 
depending  on  the  amount,  arrangement,  and  capacity  for  in- 
duction of  the  soft  iron  of  the  ship,  and  the  last  three  on  the 
amount  and  arrangement  of  the  hard  iron. 

The  parameters  a,  ~b,  c,  d,  e,  f,  g,  h,  and  Je  are  ratios  though 
physically  represented  by  rods.  They  are  not  forces  but 
become  forces  only  when  multiplied  by  the  inducing  forces 
acting  on  the  rods  they  represent.  Thus  the  force  induced  in 
the  rod  a  by  the  inducing  force  X  is  aX.  A  distinction  must 
be  drawn  between  the  case  in  which  the  coefficient  of  a  rod  is 
'+,  and  that  in  which  it  is  ( — ) . 

These  parameters  are  ratios  between  two  forces  and  their 
signs  depend  upon  the  signs  of  the  two  forces.  The  components 
of  the  earth's  force  in  the  axes  are  taken  as  the  inducing  forces 
in  those  axes,  and  since  the  cosines  of  angles  between  90°  and 
270°  are  negative,  the  inducing  force  in  any  axis  is  negative 
if  the  azimuth  of  that  axis  is  between  90°  and  270°.  Thus, 
if  the  ship  heads  NE.,  the  inducing  force  from  starboard  is 
H  cos  (90°  +  z)  =  —  .707£T,  or  a  force  equal  to  .707H"  acts 
from  the  port  beam.  Now  the  force  induced  in  a  rod  is  posi- 
tive if  it  draws  the  North  point  of  the  needle  to  head,  to  star- 
'  board,  or  vertically  downward. 

The  following  cases  will  serve  to  illustrate  the  points  dis- 
cussed. In  Fig.  44,  the  ship  heading  North,  the  inducing 


PLATE  III. 


o 

-b" 


VJ 


+(7 


( 


DIAGRAM  SHOWING  THE  POSITIONS  OP  THE  NINE  SOFT  IRON 
ROBS  WHICH  REPRESENT  THE  WHOLE  OF  THE  SOFT  IRON 
OF  A  SHIP  AS  REGARDS  ITS  ACTION  ON  THE  COMPASS. 


114 


NAVIGATION — COMPASS  DEVIATION 


force  X  from  head  is  positive ;  the  near  end  of  the  rod  a  has 
South  polarity  and  draws  the  North  end  of  the  needle  to  head 
and  the  force  induced  aX  is  positive,  therefore  the  ratio 

=    ,    v  .=  +  a»      In  Fig.  45,  the  rod  is  acted  upon  by  a 

negative  force  from  ahead  (H  cos  180°  =  X,  or  E  =( — )Z), 
or  the  inducing  force  acts  from  the  stern ;  the  near  end  of  the 
rod  has  North  polarity  and  repels  the  North  end  of  the  needle 
to  the  stern  and  the  force  induced  aX  is  negative,  therefore 
—  aX 


the  ratio  = 
seen  to  be  +  a. 


=  -\-  a.    The  coefficient  of  this  rod  is  thus 


FIG.  44. 

In  the  case  of  an  a  rod  extending  continuously  through  the 
compass,  the  signs  of  the  induced  forces  would  be  the  reverse 
of  the  above,  whilst  the  signs  of  the  inducing  forces  would  be 
unaltered,  and  the  ratio  in  both  cases  would  be  —  a. 

In  Fig.  46,  the  ship  heads  NE.,  and  the  azimuth  of  the  rod  e 
is  SE.  The  inducing  force,  (H  cos  (90°  +  z)  =  F),  is 
—  Y,  and  the  starboard  end  has  South  magnetism  induced  in 
it.  The  force  induced  draws  the  needle  to  starboard  and  is 

•4-eY 

-\-  eY,  therefore  the  ratio  =      —    =  —  e.     In  Fig.  47,  the 


ship  heads  NW.     The  inducing  force  is  +  Y.     The  starboard 
end  of  the  rod  has  induced  in  it  North  polarity  which  repels 


THEORY  OP  DEVIATIONS  OF  COMPASS  115 


the  North  point  of  the  needle,  the  induced  force  is  —  eY  '  ,  the 


_ 
ratio  =    ,     Y  =  —  e-     ^ne  coefficient  of  an  athwartship  rod 

extending  on  both  sides  of  the  compass  is  thus  seen  to  be 
-  e.  In  the  case  of  an  e  rod  entirely  on  one  side  of  the 
compass  the  induced  forces  would  have  exactly  the  opposite 
effect,  whilst  the  inducing  forces  would  remain  unchanged  in 
sign  and  the  ratio  would  be  -f-  e. 

The  signs  of  the  coefficients  of  any  of  the  remaining  rods 
may,  in  a  similar  way,  be  proven  to  be  as  indicated  in  Plate  III, 
remembering  that  it  is  convenient  to  consider  the  action  on  the 


N.W 
N.E. 


FIG.  46.  FIG.  47. 

North  point  of  the  needle  of  only  the  nearer  end  of  a  rod 
which  is  represented  as  entirely  on  one  side  of  the  compass, 
but  that  in  the  cases  in  which  the  rod  continuously  extends 
from  one  side  of  the  compass  to  the  other,  as  in  the  cases  of 
—  af  —  e,  and  —  Tcf  either  or  both  ends  may  be  considered 
as  acting. 

Five  of  the  rods,  a,  c,  e,  g,  and  Tcf  are  symmetrically  placed 
with  reference  to  the  fore-and-aft  midship  section;  &,  d,  -f, 
and  h  are  not  so  placed. 

If  the  soft  iron  of  a  ship  may  be  supposed  symmetrically 
distributed  with  reference  to  that  plane,  ~b,  d,  f,  and  h  may 
each  be  considered  equal  to  zero. 


116  NAVIGATION — COMPASS  DEVIATION 

SECTION  II. 

75.  Transformation   of   the   fundamental   equations. — To 

adapt  Poisson's  equations  to  the  various  uses  to  which  a  navi- 
gator may  apply  them,  they  must  first  be  expressed  in  terms 
of  quantities  usually  given  or  desired,  that  is,  in  terms  of 

H,  the  horizontal  force  of  the  earth ; 

H',  the  horizontal  force  of  the  earth  and  ship ; 

0,  the  magnetic  dip ; 

z,  the  magnetic  course,  or  the  azimuth  of  the  ship's  head 
measured  eastward  from  the  correct  magnetic  North  ; 

zf,  the  compass  course,  or  the  azimuth  of  the  ship's  head 
measured  eastward  from  the  direction  of  the  com- 
pass North; 

S  =  z  ~  z',  the  deviation  of  the  compass  due  to  the 
compass  heading  z  or  the  magnetic  heading  z. 

Eeferring  again  to  Fig.  43,  if  m  is  the  resultant  pole  of  the 
ship's  magnetic  force,  attracting  the  North  end  of  the  needle, 
Od  will  be  the  force  in  the  horizontal  plane  through  the 
compass,  and  the  components  of  Om  in  the  three  axes  X,  Y, 
and  Z  respectively  will  be  Oa,  Ob,  and  Of.  The  earth's  direc- 
tive force  being  OM,  the  ship's  disturbing  force  Od,  both  in 
the  horizontal  plane,  the  needle  will  be  acted  upon  by  the 
resultant  of  these,  Or  =  H',  and  the  components  of  Or  in  the 
axes  of  X  and  Y  respectively  will  be  X'  =  On  and  Y'  —  nr. 
(For  z'  in  Fig.  43,  nr  is  to  port,  and  hence  is  negative.) 

Bearing  in  mind  the  definitions  previously  given,  we  have : 

X  =  H  cos  %, 

X'  =  H'  cos  z', 

Y  =H  cos  (90°  +z)  =  —  H  sin  z, 

Y'  =  II'  cos  (90°  +  *0  =  —  H'  sin  zr, 

Z  —  H  tan  0. 

Substituting  these  values  in  equations    (17),    (18),   and 


THEORY  OF  DEVIATIONS  or  COMPASS  117 


(19),  dividing  (17)  and  (18)  by  H  and  (19)  by  Z,  we  have: 

TTl  r> 

Sr  cos  z'  =  (1  +  a)  cos  z  —  &  sin  z  +  c  tan  0  +  —•    (20) 

XI  XI 

—  ^  sin  z'  =  d  cos  3  —  (1  +  e)  sin  z  +  f  tan  0  +  £-(21) 
H  H 

~  =  ?£-;;  cos  z  -  -A-  sin  2  +  1  +  £  +  ~.  (22) 

^       tan0  tan0  Z 

Since  .ZT  is  the  force  of  the  earth  and  ship  in  the  direction 
of  the  disturbed  needle,  and  since  a  force  can  be  resolved  into 
two  components,  in  axes  at  right  angles,  all  in  the  same  plane, 
by  multiplying  it  by  the  cosine  of  the  angle  its  direction  makes 
with  each  axis,  and  since  the  denominator  in  each  equation 
denotes  the  unit  of  measure,  equation  (20)  gives  the  force  of 
earth  and  ship  to  head,  equation  (21)  the  force  of  earth  and 
ship  to  starboard,  each  in  terms  of  the  earth's  horizontal  force 
as  unit;  equation  (22)  the  force  of  earth  and  ship  downward, 
in  terms  of  the  earth's  vertical  force  as  unit.  As  the  azimuth 
of  the  ship's  head  may  be  anything  from  0°  to  360°,  it  is  de- 
sirable to  have  these  forces  in  two  fixed  directions,  —  in  the 
magnetic  meridian  and  at  right  angles  to  the  meridian. 

76.  Force  of  earth  and  ship  to  magnetic  North.  —  To  find 
the  components,  in  the  direction  of  magnetic  North,  of  the 
forces  of  earth  and  ship  acting  to  head  and  to  starboard,  mul- 
tiply (20)  by  cos  z;  and  (21)  by  cos  (90°  +  z),  or  what 
amounts  to  the  same  thing,  by  (  —  sin  z)  ;  and  take  the  alge- 
braic sum  of  the  results.  This  will  give  the  force  of  earth 
and  ship  to  magnetic  North. 

Performing  the  operations  indicated,  we  have  : 

TTf 

JT  cos  z'  cos  z  =  (1  +  a)  cos2  z  —  &  sin  z  cos  z 

p 
-f-  c  tan  6  cos  z  +  -^-  cos  z. 

Also, 

TTt 

_  sin  z'  sin  z  =  —  d  cos  z  sin  z 

£1 

+  (1  +  e)  sin2  z  —  f  tan  6  sin  z  —  -?  sin  z. 


118  NAVIGATION  —  COMPASS  DEVIATION 

And  by  algebraic  addition, 

TTt 

j^  (cos  z'  cos  z  +  sin  z'  sin  z)  =  —  (d  +  6)  sin  z  cos  z 
+  (1  +  a)  cos2  z  +  (1  +  e)   sin2  z 

+  (c  tan  0  +  TT  ]cos  2!  —(/tan  0  -f  ^  J  sin  z. 

From  plane  trigonometry, 

cos  z'  cos  z  -f-  sin  z'  sin  2  =  cos  (z  ~  z')  =  cos  8, 

cos<  ,  =  1  +  °os  ^  sin2  z  =  1-?ga' 


and  sin  2  cos  *  = 


A 

Therefore,  by  substitution, 
lcoB*  =  -  (d  +  *)  5***  +  (1  +  a) 


Collecting  the  terms  involving  the  same  function  of  the 
azimuth  z,  we  have  the  force  of  earth  and  ship  to  magnetic 
North  in  terms  of  the  earth's  horizontal  force  as  unit,  expressed 
by  the  equation, 

^cos  (5  =  1  +  —  ^—  +  (c  tan  0  +  ~)  cos  z 

-  (/tan  0  +  ^  sin  0  +  5LzJ  cos  2^  (23) 

\  -"/  * 


. 
So 

Force  of  earth  and  ship  to  magnetic  East,  and  hence  of  ship 
alone,  as  the  force  of  earth  to  East  is  always  zero.  —  In  a  simi- 
lar way,  we  may  find  the  components  in  the  direction  of  mag- 
netic East  of  the  forces  of  earth  and  ship  acting  to  head  and  to 
starboard,  multiplying  (20)  by  cos  (90°  —  z)  =  sin  z,  and 
(21)  by  cos  z,  and  adding  the  results  algebraically. 


THEORY  OF  DEVIATIONS  OF  COMPASS  119 

Performing  these  operations,  making  trigonometric  substi- 
tutions and  collecting  the  terms  as  before,  we  have  the  force 
of  the  ship  alone  to  magnetic  East  in  terms  of  the  earth's  hori- 
zontal force  as  unit,  expressed  by  the  equation, 

l^sin  d  =  fL_l  +  (c  tan  0  +  ?\  sin  z  +  //tan  0  +  ^\  cos  z 
a~e  sin  2  z  +  d  +  bcos  2z.  (24) 


If  observations  be  made  on  four  or  more  equidistant  azi- 
muths, the  mean  of  results  from  (23)  will  be  the  mean  value 
of  the  force  of  earth  and  ship  to  magnetic  North;  and  the 
mean  of  results  from  (24)  will  be  the  mean  value  of  the 
force  of  the  ship  to  magnetic  East. 

The  mean  values  from  each  equation  will  be  represented  by 
the  constant  terms,  since  the  sum  of  two,  four,  or  more  equi- 
distant values  of  sine  or  cosine  is  zero,  and  terms  having  sine 
or  cosine  as  a  factor  will  disappear. 

Therefore,  the  mean  value  of  the  force  of  earth  and  ship  to 

magnetic  North  in  terms  of  H  as  unit  is  1  +  —  s  —  5  this 

quantity,  called  A,  is  generally  less  than  unity  in  iron  ships,  a 
fact  which  indicates  a  mean  directive  force  on  the  needle  less 
than  the  earth's  directive  force  in  such  ships. 

The  mean  value  of  the  force  of  ship  to  magnetic  East  in 

terms  of  H  as  unit  is     -~   »     This  quantity  is  +  when  the 

easterly  deviations  are  in  excess  of  the  westerly  deviations, 
(  —  )  when  the  westerly  deviations  are  in  excess.  This  term 
reduces  to  zero  when  the  soft  iron  of  the  ship  is  symmetrically 
distributed  with  reference  to  the  fore-and-aft  vertical  section 
through  .the  compass. 

Letting  1  +  2-+J  be  represented  by  X,  and  dividing  (23) 


120  NAVIGATION  —  COMPASS  DEVIATION 

and  (24:)  by  A,  we  have, 


and 


+      /tan  0  +         cos  2  +          -sin  2z  (26) 

,  1  d+  I 
+  j  —  o—  cos  2  z. 

\Hj  which  is  the  mean  value  of  H'  cos  8,  or  of  the  force  of 
earth  and  ship  to  magnetic  North,,  will  be  referred  to  hereafter 
as  "  the  mean  force  to  North." 

To  simplify  these  expressions,  let  their  constant  terms  and 
the  quantities  connected  with  the  various  functions  of  the 
magnetic  course  be  represented  by  the  old  English  capital  let- 
ters as  follows  • 


=  *;     1  (/tan, 


Then  by  substituting  these  in  (25)  and  (26),  we  have: 

TJI 

Yg  cos  8  =  1  +  $8  cos  z  —  (£  sin  z  +  ©  cos  2z 

—  (5  sin  2z. 
&-  sin  8  =  51  +  '93  sin  z  +  (£  cos  «  +  3)  sin  2^ 


+  @  cos  22. 

Equation  (27)  gives  the  combined  force  of  earth  and  ship 
to  magnetic  North  and  equation  (28)  the  force  of  the  ship  to 
magnetic  East,  both  in  terms  of  the  "  mean  force  to  North  " 
as  unit  of  measure.  The  mean  value  of  (28)  is  ST,  the  mag- 
nitude and  sign  of  which  will  indicate  the  excess  of  easterly 
over  westerly  deviations,  or  the  reverse. 


THEORY  OF  DEVIATIONS  OF  COMPASS  121 

77.  Formulae  for  computing  the  deviations. — Dividing  (28) 
by  (27),  we  obtain 

3T  +  33  sin  z  +  <£  cos  z  +  S)  sin  2z  +  (£  cos  2z 
tan  8  = L _J _J ! /on } 

1  +  33  cos  s  —  (£  sin  2  +  2)  cos  2z  —  &  sin  2z    { 

which  will  give  the  deviation  (8),  by  means  of  its  tangent,  on 
any  magnetic  course  z  when  the  five  coefficients  31,  33,  (£,  2), 
and  @  are  known. 

Substituting  ^-^  for  tan  8  in  (29)  and  clearing  of  frac- 
tions, we  have : 

sin  8  -\-  3}  cos  z  sin  8  —  (£  sin  3  sin  8  +  ®  cos  22  sin  8 
—  (£  sin  22  sin  8  —  51  cos  8  +  ^3  sin  2  cos  8  +  @-  cos  z  cos  8 

+  2)  sin  2z  cos  8  +  @  cos  2s  cos  8. 
Transposing  and  collecting  terms, 
sin  8  =•  31  cos  8  +  33  (sin  z  cos  8  —  cos  z  sin  8) 

-}-  (£  (cos  2  cos  8  -\-  sin  2  sin  8)  -j-  2)  (sin  22  cos  8 
—  cos  22  sin  8)  +  ®  (cos  %z  cos  ^  +  sin  %z  s^  $)> 
but  from  plane  trigonometry, 
since  z'  =  z  —  8  and  22'  =  2  (z  —  8), 

or  2  (zf  +  ~]  =  (22  —  8)  =  (22'  +  8),  we  shall  have 

sin  8  =  31  cos  8  +  33  sin  zf  +  (£  cos  z'  +  3)  sin  (22r  +  8) 
+  (£cos  (22'  +  8). 

Equation  (30)  gives  the  deviations  by  means  of  its  sine 
nearly,  though  not  entirely,  in  terms  of  the  compass  course  z'. 

When  the  deviations  are  of  moderate  amount,  say  not  more 
than  20°,  equation  (30)  may  be  written  with  sufficient  ac- 
curacy, 

8  =  A  +  B  sin  z9  +  C  cos  zf  +  D  sin  2z'  +  E  cos  2z'  (31) 
in  which  the  approximate  coefficients,  A,  B,  C,  D,  and  E  may 
be  given  in  degrees  and  minutes  and  the  deviation  (8)  deter- 


122  NAVIGATION  —  COMPASS  DEVIATION 

mined  for  the  various  compass  courses  in  degrees  and  min- 
utes, without  introducing  a  greater  error  than  25'  in  the  com- 
puted deviation,  even  when  near  the  agreed  upon  maximum 
limit  of  20°. 

In  formula  (30),  the  coefficients  Sf,  SB,  <£,  2),  and  (£,  called 
the  exact  coefficients  of  the  various  deviations  produced,  are 
in  reality  the  forces  producing  those  deviations  expressed  in 
terms  of  the  "mean  force  to  North,"  (A~ff),  as  unit;  they 
are  very  nearly  the  natural  sines  of  the  angles  represented  by 
the  approximate  coefficients  A,  B,  C,  D,  and  E  of  equation 
(31),  if  8  is  of  moderate  amount! 

When  the  soft  iron  is  symmetrically  arranged  on  each  side 
of  the  fore-and-aft  vertical  plane  through  the  compass,  b  =  0, 
d  =  0,  /  =  0,  h  =  Q,  and  therefore  both  51  and  (£  reduce  to 
zero,  and  equations  (29)  and  (31)  become  respectively, 

33  sin  z  +  G£  cos  z  +  ®  si11  &&  . 
"  1  +  93  cos  z  —  (£  sin  z  +  ©  cos  2z 
and  8  =  B  sin  z'  +  G  cos  z'  +  D  sin  2s'.  (33) 

Equation  (30)  will  become,  by  developing  the  terms  in  the 
second  member, 
sin  8  =  21  cos  8  +  33  sin  z'  +  (£  cos  3'  +  3)  sin  2s'  cos  8 

+.  S)  cos  2s'  sin  8  +  (£  cos  2zf  cos  8  —  ©  sin  2z'  sin  8. 
In  this  expression,  8  being  small,  we  can  let  cos  8  equal  unity 
without  material  error,  and  the  above  becomes, 
sin  8  =  51  +  33  sin  z'  +  (£  cos  z'  +  $  sin  2z'  + 

(£  cos  2/  +  £)  cos  2/  sin  8  —  @  sin  2z'  sin  8. 

(£  is  very  small  and  sin  8  is  very  small,  therefore  the  last 
term  @:  sin  2zf  sin  8  is  very  small  and  may  be  neglected,  hence, 
by  transposing  ®  cos  2z'  sin  8  and  dividing  through  by  the 
expression  (1  —  S)  cos  2z'),  we  have  : 


sn   =  (34) 


1  The   correct   angular   measure    corresponding'   to    any    arc   value   is   obtained   by 
multiplying  said  arc  by  the  radian  57°.  3;  thus  D  =  S3   x  57°.3. 


THEORY  OF  DEVIATIONS  OF  COMPASS  123 

which  is  very  nearly  exact  and  gives  the  deviations  in  terms 
of  the  exact  coefficients  and  the  compass  courses. 

When  the  soft  iron  is  symmetrically  distributed  51  and  (£ 
will  disappear  in  (34). 

78.  Subdivisions  of  the  deviation. — In   (31),  the  several 
parts  of  the  deviation  are: 

A,  the  constant  deviation  due  to  transient  magnetism  in- 
duced in  soft  iron  represented  by  parameters  d  and  ~b, 
or  to  constant  errors  of  observation,  etc. ; 

B  sin  zr  +.  0  cos  z',  the  semicircular  deviation  due  to  sub- 
permanent  magnetism  and  the  transient  magnetism  in- 
duced in  vertical  soft  iron; 

D  sin  2z'  +  E  cos  2z',  the  quadrantal  deviation.  The  first 
or  larger  part,  D  sin  2z'9  is  due  to  transient  magnetism 
in  horizontal  soft  iron  symmetrically  situated  with 
reference  to  the  fore-and-aft  vertical  plane  through  the 
compass.  The  second,  or  smaller  part,  E  cos  2z'9  is 
due  to  transient  magnetism  induced  in  horizontal  un- 
symmetrically  situated  soft  iron. 

79.  Relation  between  A  and  £). — Both  these  coefficients  de- 
pend for  their  value  on  the  parameters  a  and  e.     The  coeffi- 
cient ®,  which  equals  -—a  "~  e ,  represents  the  force  in  terms 

A  til 

of  \H  which  produces  the  larger  part  of  the  quadrantal  devia- 
tion, the  force  itself  being  XffS).  This  part  is  generally  posi- 
tive, being  due  to  -(-  a  and  —  ef  or  the  excess  of  —  e  over  —  a. 
The  horizontal  force  at  the  compass  is  that  of  the  earth,  the 
ship's  subpermanent  magnetism,  and  that  induced  in  soft  iron 
combined  and  acting  in  the  direction  of  the  compass  needle, 
this  resultant  force  being  represented  by  H'.  The  component 
to  magnetic  North  is  H'  cos  8  and  the  mean  value  of  this  com- 
ponent for  an  entire  revolution  of  the  ship's  head  is  X5",  which 
is  generally  less  than  unity,  H  itself  being  considered  as  unity. 
In  the  complete  revolution  of  the  ship's  head,  the  increase  of 


124 


NAVIGATION — COMPASS  DEVIATION 


directive  force  due  to  the  subpermanent  magnetism  in  one 
semicircle  exactly  equals  the  decrease  due  to  the  same  cause  in 
the  other  semicircle,  so  the  diminution  in  the  value  of  XH 
is  not  due  to  subpermanent  magnetism  but  arises  from  hori- 
zontal induction  in  soft  iron  represented  as  constituent  parts 
of  A,  therefore  A  is  generally  less  than  unity  and  as  it  equals 

1  +  a  T"  e ,  the  diminution  is  due  to  —  a  and  —  e,  or  the 

<i 

excess  of  —  e  over  +  a. 

Since  S)  is  usually  positive,  and  A  is  usually  less  than  unity, 
it  is  to  be  inferred  that  the  usual  distribution  of  horizontal 
soft  iron  symmetrically  situated  may  be  represented  by  the 
arrangement  of  Fig.  48,  or  Fig.  49  in  which  the  +  2)  would 
be  due  to  the  excess  of  effect  of  —  e  over  —  a. 


-a 


+  e 


+e 


FIG.  48. 


FIG.  49. 


FIG.  50. 


FIG.  51. 


An  arrangement  of  soft  iron  as  shown  by  Fig.  50,  in  which 
r-\-  a  =  +  e,  would  give  A  a  value  greater  than  unity,  or  in- 
crease the  "  mean  force  to  North  ";  that  as  shown  by  Fig.  51, 
in  which  —  a  =  —  e,  would  give  A  a  value  less  than  unity,  or 
diminish  the  "  mean  force  to  North  " ;  but  neither  would  cause 
any  deviation. 

When  ®  is  caused  by  an  arrangement  of  iron  represented  by 
parameters  of  one  sign,  it  may  be  neutralized  by  compensators 
producing  the  effect  of  parameters  of  the  opposite  sign. 

We  may  find  a  and  e  from  A  and  3)  thus : 
1      a- 

—  .'.  z/.  M)  r=  ti  —  «     I 

.  — 1.  (35) 


=1+ 


=A(l— $)_i.  (36) 


THEORY  OF  DEVIATIONS  OF  COMPASS  125 

80.  Relation  between  21  and  (£ — These  are  the  forces,  in 
terms  of  XH,  which  cause  respectively  the  constant  deviation 
and  the  smaller  part  of  the  quadrantal  deviation,  and  since 

$  =  1^-^and  @  =  i  d  +  l,  they  are  closely  related;  the 

/I          A  A          & 

forces  causing  these  deviations  arise  from  transient  magnetism 
induced  in  horizontal  soft  iron  unsymmetrically  situated  with 
reference  to  the  fore-and-aft  vertical  plane  through  the  com- 


They  both  reduce  to  zero  when  the  .soft  iron  is  symmetrically 
distributed  about  the  compass. 

51  is  marked  +  when  the  easterly  deviation  is  in  excess, 
otherwise  ( — ). 

The  effect  of  (£  is  very  small  and  is  marked  +  when  it  causes 
easterly  deviations  between  North  and  NE. 

81.  The  coefficient  23. — The  fore-and-aft  component  of  the 
magnetic  force  which  causes  semicircular  deviation  is  XH"33, 
and  S3  represents  that  component  if  expressed  in  terms  of  \H 
as  unit,  in  other  words  S3  is  the  ship's  polar  force  to  head  in 
terms  of  the  "  mean  force  to  North  "  as  unit.  It  is  marked  + 
if  drawing  the  North  point  of  the  needle  to  head,  otherwise 

(-)• 

Since  S3  =  i  (c  tan  0  +  ~\  =  -^-  (cZ  +  P),  it  is  seen 

that  the  ship's  polar  force  to  head,  expressed  in  terms  of 
"mean  force  to  North"  as  unit,  is  composed  of  (1)  the  fore- 
and-aft  component  of  the  ship's  subpermanent  magnetism  P, 
+  if  acting  to  head,  ( — )  if  to  the  stern;  (2)  the  fore-and-aft 
component  of  the  force  due  to  transient  magnetism  induced 
in  vertical  soft  iron,  or  cZ,  +.  if  acting  to  head,  ( — )  if  to 
the  stern. 

That  part  due  to  subpermanent  magnetism  varies  inversely 
as  H",  and  that  part  due  to  vertical  induction  in  vertical  soft 
iron  varies  directly  as  the  tangent  of  the  magnetic  dip. 


126  NAVIGATION — COMPASS  DEVIATION 

The  force  23  may  be  counteracted  by  a  fore-and-aft  magnet 
with  its  center  in  a  transverse  vertical  plane  passing  through 
the  compass,  so  placed  that  the  North  end  is  forward  if 
23  is  +,  aft  if  93  is  (— ). 

Eegarding  21  and  (£  as  zero,  equation  (32)  shows  23  to  be 
the  only  force  causing  deviation  when  the  ship  heads  magnetic 
East  or  West,  at  which  time  it  exerts  almost  its  maximum 
effect;  therefore  to  neutralize  23,  head  the  ship  magnetic 
East  or  West,  and  move  a  fore-and-aft  system  of  magnets, 
placed  as  in  the  preceding  paragraph,  towards  or  from  the 
compass  till  the  compass  heading  is  East,  or  West,  as  the  case 
may  be. 

82.  The  coefficient  (£. — The  athwartship  component  of  the 
magnetic  force  which  causes  semicircular  deviation  is  Afl"(£, 
and  &  represents  that  component  if  expressed  in  terms  of  XH 
as  unit,  in  other  words  &  is  the  ship's  polar  force  to  starboard 
in  terms  of  the  "  mean  force  to  North  "  as  unit.  It  is  marked 
+  if  drawing  the  North  point  of  the  needle  to  starboard,  ( — ) 
if  to  port. 

Since  £  =  ~  t  f  tan  0  +  ~S\  =  — L-  (fZ  +  Q),  it  is  seen 

that  the  ship's  polar  force  to  starboard,  expressed  in  terms  of 
the  "  mean  force  to  North  "  as  unit,  is  composed  of  ( 1 )  the 
athwartship  component  of  the  ship's  subpermanent  mag- 
netism Q,  +  if  acting  to  s'tarboard,  ( — )  if  acting  to  port; 
(2)  the  athwartship  component  of  the  force  due  to  transient 
magnetism  induced  in  vertical  soft  iron,  or  fZ9  +  if  acting 
to  starboard,  ( — )  if  to  port. 

That  part  due  to  subpermanent  magnetism  varies  inversely 
as  //,  and  that  part  due  to  vertical  induction  in  vertical  soft 
iron  varies  directly  as  the  tangent  of  the  magnetic  dip. 

The  force  (£  may  be  counteracted  by  an  athwartship  magnet 
with  its  center  in  a  fore-and-aft  vertical  plane  passing  through 
the  compass,  so  placed  that  the  North  end  is  to  starboard  if 
(£  is  +,  to  port  if  (£  is  ( — ) . 


THEORY  OF  DEVIATIONS  OF  COMPASS  127 

Eegarding  St  and  (£  as  zero,  equation  (32)  shows  (£  to  be 
the  only  force  causing  deviation  when  the  ship  heads  magnetic 
North  or  South,  at  which  time  it  exerts  almost  its  maximum 
effect;  therefore  to  neutralize  (£,  head  the  ship  magnetic 
North  or  South,  and  move  an  athwartship  system  of  magnets, 
placed  as  in  the  preceding  paragraph,  towards  or  from  the 
compass  till  the  compass  heading  is  North,  or  South,  as  the 
case  may  be. 

83.  The  ship's  polar  force  and  the  starboard  angle  a.  — 
Both  XH%$  and  \H&,  in  other  words  their  resultant 


\-  (£2,  the  ship's  horizontal  polar  force,  may  be  cor- 
rected by  a  single  magnet,  or  a  system  of  parallel  magnets, 
whose  center  is  immediately  below  the  compass  center  and 
whose  axis  is  horizontal  and  makes  an  angle  a  with  the  fore- 

and-aft  line  through  the  compass.     This  angle  =  tan'1  ^; 

it  is  called  the  starboard  angle,  and  is  measured  from  ahead 
around  to  the  right.     In  other  words,  there  is  a  magnetic  force 


+  @-2  drawing  the  North  point  of  the  needle  toward  a 
fixed  point  in  the  ship  of  South  polarity,  the  direction  of  which 
with  the  fore-and-aft  line,  measured  as  above,  is  a.  In  com- 
pensation that  field  of  South  polarity  must  be  neutralized  by 
one  of  North  polarity. 

84.  Variations  in  the  parameters:  The  Gaussin  error.  — 
Poisson's  equations  are  based  on  the  hypothesis  that  the  mag- 
netism of  a  ship  is  partly  permanent  and  partly  transient, 
that  in  consequence  all  the  parameters  are  constant  and  all  the 
exact  coefficients,  except  93  and  (£,  are  constant;  the  change 
in  93  and  (£  taking  place  only  on  change  'of  magnetic  latitude, 
and  then  because  they  involve  the  dip  and  horizontal  force. 
The  hypothesis  is  not  absolutely  true,  the  magnetism  due  to 
hard  iron  is  only  subpermanent,  and  the  transient  magnetism 
is  never  that  due  to  the  inducing  force  at  the  time  and  place 
but  that  due  to  a  force  of  some  previous  time  and  place;  in 


128  NAVIGATION — COMPASS  DEVIATION 

other  words  there  is  a  retardation  in  induction,,  and  this  occurs 
whether  the  ship  is  on  a  straight  course  or  is  turning  in  a 
circle. 

These  effects  may  be  considered  as  unimportant  in  all  soft 
iron  except  the  parameters  a,  e,  and  g,  which  are  not  only 
comparatively  large  in  amount,  but  cut  the  lines  of  force  at 
all  angles  as  the  ship  swings  in  azimuth,  so  that  any  slowness 
in  receiving  or  parting  with  a  full  charge  of  magnetism  has 
an  important  bearing  on  the  deviation. 

Take  the  rod  ( — )  e,  for  example,  in  the  case  of  a  vessel 
swinging  with  starboard  helm  from  a  heading  East;  its  port 
end  having  come  from  the  North  repels,  and  its  starboard  end 
having  come  from  the  South  attracts  the  North  end  of  the 
needle,  even  when  the  rod  is  at  right  angles  to  the  line  of 
force  and  when,  by  the  hypothesis,  it  should  fail  of  effect. 
The  force  due  to  this  lagging  of  the  magnetism  causes  a  +  91 
and  a .+!  @  when  the  ship  swings  to  the  left,  a  ( — )  SI  and 
a  ( — )  (£  when  it  swings  to  the  right.  This  error  is  known 
as  the  "  Gaussin  error "  after  M.  Gaussin  who  first  called 
attention  to  it.  For  accuracy  a  ship  should  either  be  swung 
with  both  helms  and  a  mean  of  the  two  deviations  taken;  or 
swung  very  slowly  with  one  helm,  the  ship  being  kept  steady 
on  each  rhumb  at  least  four  minutes.  When  only  one  helm  has 
been  used,  it  is  proper  to  leave  uncorrected  at  North  and  South 
a  little  easterly  error  if  ship  was  swung  to  the  left,  a  little 
westerly  error  if  swung  to  the  right,  in  all  cases  when  the  ship 
was  not  swung  very  slowly. 

SECTION  III. 

85.  Determination  of  the  coefficients.  Method  of  least 
squares. — It  has  been  shown  that  the  deviation  (if  of  moder- 
ate amount,  say  not  exceeding  20°),  may  be  expressed  for 
any  heading  z'  per  compass  by  the  formula 

8  =  A  +  B  sin  z'  +  C  cos  z'  +  D  sin  2z'  +  E  cos  2z', 
which  contains  five  unknown  quantities. 


THEORY  OF  DEVIATIONS  OF  COMPASS  129 

If  the  deviation  is  observed  on  five  different  headings, 
these  coefficients  may  be  determined  by  elimination  from  the 
resulting  equations;  but,  if  observations  are  made  on  more 
than  five  headings,  there  will  be  more  than  five  equations,  and 
the  values  of  the  unknown  coefficients  found  from  any  five  of 
them  will  in  general  not  satisfy  the  others  by  an  amount 
called  the  error  which  may  be  +  or  ( — ) . 

The  greater  the  number  of  observations  to  determine  an 
unknown  quantity,  the  more  accurate  will  be  its  value,  pro- 
vided the  observations  are  all  carefully  taken  and  are  equally 
trustworthy ;  therefore  if  we  can  determine  the  five  coefficients 
from  observations  on  as  many  as  24  headings  so  much  the 
better,  and,  if  not  on  24,  then  on  as  many  as  possible.  The 
number  of  the  observation  equations  being  greater  than  the 
number  of  the  unknown  quantities,  they  must  be  so  adjusted 
as  to  give  the  most  probable  values  of  the  five  quantities 
sought. 

The  "Theory  of  Probability  of  Errors"  proves  that  the 
most  probable  values  of  unknown  quantities  are  those  that 
reduce  the  sum  of  the  squares  of  the  errors  to  a  minimum. 

Suppose  the  observations,  greater  in  number  than  the  num- 
ber of  the  unknown  quantities,  give  rise  to  equations  of  con- 
dition of  the  form 


ci  y  +  d\  z  +  •  •  •  •  =  0 
C2  y  +  d2  z  +  •  •  •  -  =  0 

etc.,  etc., 


(37) 


in  which  a± ,  a2  ,  a3 ,  6X ,  &2 ,  &3 ,  etc.,  are  known  quantities 
and  x,  y,  z,  etc.,  are  unknown  quantities. 

When  the  most  probable  values  of  the  unknown  quantities 
are  substituted  in  group  (37),  none  of  the  equations  will  be 
fully  satisfied;  in  other  words  each  will  fail  to  reduce  to  zero 


130  NAVIGATION — COMPASS  DEVIATION 

by  a  small  error;  E±  in  the  first,  E2  in  the  second,  and  so  on, 
and  the  equations  will  become 

a2  +  12  x  +  c2  y  +  d2  z  +  .  .  .  .  =  E2     I  . 

etc.,  etc., 

and  the  sum  of  the  squares  of  the  errors  will  be  a  minimum. 
To  find  the  most  probable  value  of  x  from  group  (38),  it 
is  only  necessary  to  consider  x;  so  the  terms  independent  of  x 
may  be  represented  in  the  left  hand  member  of  each  equation 
in  both  .groups  (37)  and  (38)  by  M x ,  M 2 ,  M3 ,  etc.,  respect- 
ively, and  we  will  have  from  (38), 

M4-^=^l 

?ti5z5 1  (39) 

t>3x  +  Ms  =  Es 

etc.,  etc. 
Squaring  both  members  of  each  equation  and  adding,  we  have 


To  find  the  most  probable  value  of  x,  we  must  make  the -sum 
of  the  squares  of  the  errors  a  minimum,  or,  what  amounts  to 
the  same  thing,  make  the  left  hand  member  of  equation  (40) 
a  minimum. 

Differentiating  it  with  respect  to  x  and  placing  the  first 
differential  equal  to  zero,  after  dividing  through  by  the  com- 
mon factor  2  dx,  we  have 

(6,  x  +  M,)  I,  +  (&2  x  +  M2)  Z>2 

+  (6.  x  +  M3)  I,  +  . . .  =  0 
which  gives  the  most  probable  value  of  x. 

The  equation  (41)  is  called  the  normal  equation  in  x  and 
equals  the  sum  of  the  equations  in  group  (37)  taken  after  each 
equation  has  been  multiplied  by  the  coefficient  of  x  in  that 
equation. 


THEORY  OF  DEVIATIONS  OF  COMPASS  131 

In  the  same  way,  letting  Ni9  N2  ,  Ns  ,  etc.,  represent  in 
the  equations  of  group  (38)  all  the  terms  independent  of  y, 
we  will  have  the  normal  equation  in  y 


(c2y  +  N2)  c2 


which  gives  the  most  probable  value  of  y.  This  normal  in  y 
equals  the  sum  of  the  equations  in  group  (37)  taken  after 
each  equation  has  been  multiplied  by  the  coefficient  of  y  in 
that  equation. 

Similarly,  we  may  find  the  normal  equation  in  z,  and  in  all 
the  other  unknown  quantities. 

The  normal  equations  will  be  the  same  in  number  as  the 
unknown  quantities,  and  the  value  of  these  quantities  ob- 
tained therefrom  will  be  the  most  probable  value. 

Comparing  the  normal  equations  with  the  equations  of  con- 
dition, the  following  rule  for  the  formation  of  the  normals 
is  evident. 

"Multiply  each  equation  of  condition  by  the  coefficient  of 
each  unknown  quantity  in  that  equation  taken  with  its  sign. 
The  sum  of  the  resulting  equations  in  which  the  coefficients  of 
x  are  multipliers  will  be  the  normal  equation  in  x}  and  simi- 
larly for  the  others/' 

86.  Let  80  ,  8±  ,  82  ,  etc.,  be  the  deviations  obtained  from 
observations  when  the  ship  heads  on  the  15°  rhumbs  per  com- 
pass represented  by  z'0,  z\,  z'2,  etc.,  then  (31)  will  become 

sin  z'0-\-C  cos  z'0+D  sin  2z'0+E  cos  2z'0* 
sin  z\+C  cos  z\+D  sin  2z\+E  cos  2z\ 
82=A+B  sin  z'2+C  cos  z'2+D  sin  2z'2+E  cos  2z'2 

.(43) 


=A+#  sin  z'23+C*cos  z'23+D  sin  2z'2S+E  cos  2z'2 


132 


NAVIGATION — COMPASS  DEVIATION 


In  the  above  the  sines  and  cosines  may  be  replaced  by  the 
symbols  S0,  Slf  S2,  etc.,  representing  the  sines  of  angles  of 
0°,  15°,  30°,  etc.,  remembering,  however,  that  the  cosine 
of  an  angle  is  the  sine  of  its  complement,  that  is,  cos  z± 
=  siii  z5,  cos  z2  =  sin  z4,  etc.,  also  that  sin  2z^  =  sin  z2, 
or  sin  2zt  =  82,  etc. 

For  uniformity's  sake,  the  symbols  8Q  and  86  which  are 
respectively  zero  and  unity  will  appear  as  multipliers. 

Strict  attention  must  be  paid  to  the  signs  of  the  functions 
on  the  azimuths  used. 

Making  the  proper  substitutions  we  have 

80  =  A  +  B80  +  C8Q  +  D80  +  E8Q 
8±  =  A.  +  BSi  +  C85  +.D82  +  E8, 
S2  =  A+BS2  +  C8±  +  D8,  +  ES2 


(44) 


823  =  A  —  B8i  +  C85  —  DS2  +  E8 

which  are  the  24  observation  equations,  or  equations  of  con- 
dition, from  which  the  five  normal  equations  must  be  found 
by  the  method  of  least  squares  as  expressed  in  the  rule  of 
Article  85,  before  the  coefficients  A,  B,  C,  D,  and  E  can  be 
found. 

To  form  the  normal  equation  in  A. — Multiply  each  equa- 
tion of  group  (44)  by  the  coefficient  of -A  in  it;  the  coefficient 
in  each  case  being  unity,  no  change  will  be  made  in  any 
term.  Then  adding  all  members  on  each  side  of  the  equality 
sign,  we  shall  have. 


8,,  =  24A. 


(45) 


Since  the  sines  and  cosines  of  courses  differing  by  180°  are 
numerically  the  same  with  opposite  signs,  their  sum  is  zero, 


THEORY  OF  DEVIATIONS  OF  COMPASS  133 

and  the  summation  of  sines,  or  cosines,  on  an  even  number 
of  equidistant  azimuths  is  zero;  hence  terms  involving  B,  C, 
D,  and  E  do  not  appear  in  (45). 

Equation  (45)  may  be  put  in  the  following  form  : 


*  (*9  +  *2l)   >~   f  =  6  ^,  <46> 

+  *   U  («4  +  «1G>  +  i  («10+  ««)   I"   +  H  *  («5  +  517>  +  *  («11+  S23)  }"  J 

which  explains  the  formation  of  columns  (5)  and  (11)  of  the 
analysis  sheet,  Article  89,  and  the  steps  leading  to  the  find- 
ing of  A;  for  instance,  J(80  +  812)  will  give  the  mean  value 
of  A  and  E  on  N.  and  S.  ;  \  (86  +  818)  will  give  the  mean  value 
of  A  and  E  on  E.  and  W.  ;  the  A  will  have  the  same  sign  in 
both  cases,  the  E  will  have  the  contrary  sign  in  each  case; 
therefore,  i  -{  i(80  +  §12)  +  4(8«  +  8is)  f  will  give  the  mean 
value  of  A  on  four  rhumbs,  and  as  each  term  in  a  bracket  rep- 
resents a  mean  value  of  A  for  four  rhumbs,  we  shall  have  6  A 
on  the  opposite  side  of  the  equality  sign  in  (46). 

To  form  the  normal  in  B.  —  Multiply  each  equation  of 
group  (44)  by  the  coefficient  of  B  in  that  particular  equation, 
having  due  regard  for  the  signs  ;  in  other  words  multiply  the 
first  equation  by  S0,  the  second  by  S19  etc.,  then  add  results 
on  both  sides  of  the  equality  sign.  On  the  right  hand  side 
all  the  coefficients  except  B  will  disappear  as  the  summation 
of  the  resulting  multipliers  for  them  will  consist  of  two  parts, 
each  part  containing  identical  terms  with  opposite  signs; 
therefore, 
.S, 

=  B  \  2  (#0*  +  #62)  +  4 

=  B  -{  2  +  4  (1  +  1  +   i)  }-  =  12B 

and  the  normal  in  B  is,  for  observations  on  24  15°-rhumbs, 

80^0  +  81^1  +  82^2  ----  —  828  #1  =  125.        (47) 

The  grouping  of  the  deviations  in  the  analysis  sheet,  the 

finding  of  column   (6)   marked  semicircular  deviation,  and 


134  NAVIGATION  —  COMPASS  DEVIATION 

the  use  of  the  divisor  6  in  finding  B  from  observations  on  24 
headings  is  thus  explained;  i.  e.,  since  S0  is  multiplied  by  the 
same  quantity  numerically  as  S12,  Sx  the  same  as  813,  etc.,  but 
with  a  different  sign,  and  since  semicircular  deviation  is  the 
algebraic  difference  of  the  deviations  on  opposite  rhumbs  di- 
vided by  2,  if  the  results  in  (47)  be  grouped  according  to  the 
multipliers  and  divided  by  2,  then  (47)  will  become 

..)<Si'+i(*,^«i«)ff. 
—  »u)81  =  6B, 

which  explains  the  steps  pursued  in  the  analysis  sheet  leading 
to  the  determination  of  B. 

To  form  the  normal  equation  in  C.  —  Multiply  each  of  the 
24  equations  of  condition  of  group  (44)  by  the  coefficient  of 
C  in  its  own  equation,  and  proceed  as  in  finding  the  normal 
in  B.  All  the  coefficients,  except  C,  will  disappear  as  the 
summation  of  resulting  multipliers  for  them  will  consist  of 
two  parts,  each  part  containing  identical  terms  with  opposite 
signs,  and  we  shall  have  the  normal  equation  in  C  : 

80  S6  +  Si  S5  +  S2  £4  +  ----  823  S5  =  12  0.        (49) 
This  may  be  put  under  the  following  form  : 


!«*  .     . 

+  .  .  .  .*(«„-  8,,)  (-$,)=  6(7, 

which  explains  the  formation  and  use  of  column  (6)  of  the 
analysis  sheet  and  the  steps  leading  to  the  determination  of 
C  from  observations  on  24  equidistant  15°  rhumbs. 

To  find  the  normal  equation  in  D.  —  By  the  same  rule  and 
similar  methods,  as  in  the  cases  of  B  and  C,  we  will  obtain  the 
normal  in  D: 
80  £0  +  8±  82  +  82  St  +  83  S6  +  .  .  .  .  —  823  S2  =  12  D.      (  51  ) 

Since  the  functions  of  twice  the  azimuth  are  used,  and  since 
S0  may  be  considered  +  or  (  —  ),  the  following  will  have 
negative  multipliers,  86  up  to  81±  inclusive,  and  818  to  823  in- 


THEORY  OF  DEVIATIONS  OF  COMPASS  135 

elusive.     If  equation  (51)  is  grouped  by  multipliers  and  di- 
vided twice  by  2,  we  shall  have 

4  •{  4(*o  +  «12)  -  l(«o  +  *«)  }•  fib  +  4  •!  4  (»i  +  813)  -  4(S7  +  «19)  }•  $f| 
+  4 •{  4(«2  +  814)  - 4(88  +  820)  }•  fi*  +  4  •!  4  (8s  +  «i6)  -  4(«9  +  «n)  }•  fie  f  =  3Z>  (52) 

+  4 -I  4(84  +  5ie)  -  4(8io+  5ia)  }•  fi1*  +  4  U  (86  +  8U)  - i(5n+  *2s)  }- S2J 
which  explains  the  formation  of  columns  (5)  and  (12)  and 
the  steps  leading  to  the  determination  of  D  in  the  analysis 
sheet. 

The  normal  equation  in  E. — By  the  same  rule  and  methods 
similar  to  those  pursued  in  the  case  of  D,  we  have  the  normal 
equation  in  E : 

80^6  +  81^  +  82^2 +Z2384  =  12E.      (53) 

If  equation  (53)  is  grouped  by  multipliers  and  divided 
twice  by  2,  we  shall  have 

J-       86  +4-1  i(S1+Si3)-4(ST  +*w)  r      S*  "] 

l-        S2   +4U(83+«i6)-4(«9+«a)}-       ^o    ^=3^     (54) 

H  -  fia  K4-I  4(«6+8ll)-4(*U+*88)  }•  "I  -#4  U 

which  explains  the  steps  leading  up  to  the  determination  of 
E  in  the  analysis  sheet. 

If  observations  are  taken  on  only  12  equidistant  headings, 
the  divisors  in  the  analysis  sheet  will  be  for  A,  E,  and  Q,  3 
instead  of  6,  and  for  D  and  E,  f  instead  of  3. 

If  taken  on  only  the  eight  principal  rhumbs,  the  divisors  for 
A,  B,  and  C  will  be  2,  and  for  D  and  E  unity. 

87.  The  expression 

S  =  A  +  B  sin  z'  +  C  cos  z'  +  D  sin  2z'  +  E  cos  2«' 
considers  the  deviation  as  composed  of  only  the  constant,  semi- 
circular, and  quadrantal  deviations,  while  in  fact  terms  of  a 
sextantal'type,  octantal  type,  etc.,  may  exist. 

This  is  shown  by  the  following  equation : 
S  =  A  +  B  sin  2'  +  C  cos  z'  +  D  sin  2z'  +  E  cos  2.z' 

+  jP  sin  3z'  +  0  cos  82'  +  H  sin  4z'  +  K  cos  4s'          (55) 
+  L  sin  5s'  +  Jf  cos  5z'  +  tf  sin  6z'. 


136  NAVIGATION — COMPASS  DEVIATION 

which  is  obtained  from  (30)  by  a  series  of  expansions,  sub- 
stitutions, and  eliminations,,  and  which  is  exact  to  terms  of 
the  third  order  inclusive. 

88.  Determination  of  exact  coefficients. — For  all  practical 
purposes  the  deviation  is  expressed  with  sufficient  accuracy  by 
the  five  approximate  coefficients ;  if,  however,  greater  accuracy 
is  required,  as  when  the  deviation  much  exceeds  20°,  it  should 
be  expressed  by  the  exact  coefficients.     These  may  be  deter- 
mined by  the  method  of  least  squares,  or  from  the  following 
equations  involving  the  approximate  coefficients : 

2T  =  sin4.  (56) 

33  =  sin  B  (1  +  £  sin  D  +  TV  versin  B  —  J  versin  C) 

+  £  sin  0  X  sin  #.  (57) 

&  =  sin  C  (1  —  ^  sin  D  —  J  versin  B  -f-  TV  versin  C) 

+  4  sin  B  X  smE.  (58) 

®  =  sin  D  (1+  $  versin  D).  (59) 

(£  =  sin  E  —  sin  A  X  sin  D.  (60) 

SECTION  IV. 

89.  Analysis  of  deviations  and  the  use  of  the  form. — The 

deviations  entered  on  this  form  should  be  for  the  15°  compass 
rhumbs,  and,  if  the  observations  were  not  on  these  rhumbs, 
plot  the  deviations  on  a  Napier  diagram,  and  take  off  the 
total  deviation  on  each  15°  compass  rhumb  and  transfer  it 
to  column  2  or  4,  Table  I,  of  the  form.  The  following  ex- 
ample will  illustrate  the  process : 

Ex.  8. — Compute  the  approximate  and  exact  coefficients 
from  the  deviation  table  found  in  Ex.  4,  Art.  56.  Find  also 
the  starboard  angle  a. 

(1)  Write  down  the  observed  deviations  in  columns  (2) 
and  (4),  opposite  the  proper  rhumbs,  prefixing  the  sign  -f-  "to 
the  easterly  deviations,  the  sign  ( — )  to  the  westerly  devia- 
tions. 


THEORY  OF  DEVIATIONS  OF  COMPASS  137 

(2)  Form  column   (5)   by  taking  half  the  algebraic  sum 
of  columns  (2)  and  (4).     Since  the  constant  and  quadrantal 
deviations  have  the  same  sign  and  the  semicircular  has  the 
opposite  sign  on  azimuths  differing  180  °,  this  process  elimi- 
nates the  semicircular  deviation,  and  column  (5)  records  the 
constant   and  quadrantal   deviation   on  the   equidistant   15° 
rhumbs  per  compass  from  0°  to  165°  inclusive,  or  from  180° 
to  345°  inclusive. 

(3)  Form  column  (6)  by  taking  half  the  algebraic  differ- 
ence of  columns   (2)   and   (4)  ;  or  what  is  the  same  thing 
change  mentally  the  signs  of  the  quantities  in  column  (4), 
then  take  half  the  sum  of  columns  (2)  and  (4),  entering  re- 
sults in  column  (6).    This  process  eliminates  the  constant  and 
quadrantal  deviations,  column  (6)  being  the  semicircular  de- 
viation on  the  equidistant  15°  rhumbs  per  compass  from  0° 
to  165°  inclusive,  or  from  180°  to  345°  inclusive,  if  we  con- 
sider the  signs  changed. 

The  correctness  of  columns  (5)  and  (6)  may  be  proved  by 
adding  them  algebraically;  the  sum  of  the  quantities  opposite 
any  heading  should  equal  the  quantity  in  column  (2)  oppo- 
site the  same  heading. 

(4)  Since  the  semicircular  deviation  on  any  compass  azi- 
muth of  the  ship's  head  z'  is  B  sin  z'  +  C  cos  z'y  the  quan- 
tities in  column   (6)    are  multiplied  by  the  multipliers  set 
opposite  them  in  columns  (7)  and  (8)  to  form  respectively  the 
products  of  columns  (7)  and  (8),  the  first  set  of  multipliers 
represented  by  S0,  8lf  S2  .  .  .  .  30  being  the  natural  sines  of 
the  rhumbs  0°,  15°,  30°  ....  90°  respectively,  and  the  sec- 
ond set  represented  by  S6,  85  .  .  .  .  80  being  the  natural 
cosines  of  the  same  rhumbs,  S0  being  0  and  SQ  unity.     The 
multiplication  is  facilitated  by  Table  IV  of  this  book,  or  by 
Table  X  of  "  Diehl's  Compensation  of  the  Compass." 

When  the  angle  is  greater  than  the  tabulated  arc,  as  42°  10', 
it  may  be  divided  into  two  parts,  each  part  to  come  within  the 


138  NAVIGATION — COMPASS  DEVIATION 

limit  of  the  table,  as  30°  and  12°  10',  and'  the  sum  of  the 
results  for  the  two  parts  taken. 

In  these  multiplications  careful  attention  must  be  paid  to 
the  rule  of  signs;  that  is,  +  multiplied  by  +>  or  ( — )  multi- 
plied by  ( — )  gives  -}-,  and  +  multiplied  by  ( — )  gives  ( — ) . 

The  algebraic  sum  of  the  products  in  each  of  the  columns 
(7)  and  (8)  divided  by  6,  3,  or  2,  according  as  the  obser- 
vations were  taken  on  24,  the  12  or  8  principal  compass 
rhumbs,  will  give  from  column  (7)  the  approximate  coefficient 
B  and  from  column  (8)  the  approximate  coefficient  C. 

An  approximate  check  on  B  may  be  obtained  by  taking  the 
mean  of  the  deviations  at  East  and  at  West,  the  sign  of  the 
latter  being  changed;  an  approximate  check  on  C  may  be 
obtained  by  taking  the  mean  of  the  deviations  at  North  and 
South,  the  sign  of  the  latter  being  changed. 

(5)  Proceed  to  find  A,  V,  and  E,  following  in  Table  II  a 
process  similar  to  that  followed  in  finding  B  and  C. 

Write  down  the  upper  half  of  column  (5)  in  column  (9) 
and  the  lower  half  of  column  (5)  in  column  (10).  From 
columns  (9)  and  (10),  form  columns  (11)  and  (12)  in  the 
same  way  in  which  we  formed  columns  (5)  and  (6)  of  Table 
I,  and  prove  their  correctness  in  the  same  manner. 

It  will  be  readily  seen  that  by  this  process  we  have  separated 
the  constant  and  quadrantal  deviations;  column  (11)  is  the 
constant  part  of  the  deviation,  each  of  the  eight  values  being 
derived  from  the  deviations  on  four  rhumbs  of  the  compass  90° 
from  each  other,  being  the  mean  of  the  deviations  as  repre- 
sented in  the  brackets  of  the  left-hand  member  of  equation 
(46). 

(6)  The  sum  of  the  quantities  in  column  (11)  divided  by 
6,  3,  or  2  according  as  the  observations  were  taken  on  the  24, 
12,  or  8  principal  rhumbs  will  give  the  value  of  A.    An  ap- 
proximate check  on  A  may  be  obtained  by  taking  the  alge- 
braic mean  of  the  deviations  on  the  4  cardinal  points. 


ANALYSIS  OF  DEVIATIONS  139 

Column  (12)  is  the  quadrantal  deviation  on  15°  rhumbs 
from  0°  to  75°  and  from  180°  to  255°;  or,  with  the  signs 
changed,  the  quadrantal  deviation  from  90°  to  165°,  or  from 
270°  to  345°.  Each  of  the  eight  values  in  column  (12)  is 
derived  from  the  deviations  on  four  rhumbs  of  the  compass  90° 
apart,  as  shown  in  equations  (52)  and  (54). 

(7)  Since  the  quadrantal  deviation  on  any  compass  head- 
ing of  the  ship  zf  is  D  sin  2z'  +  E  cos  2z',  the  quantities  in 
column  (12)  are  multiplied  by  the  multipliers  set  opposite 
them  in  columns  (13)  and  (14)  to  form  respectively  the  prod- 
ucts of  columns  (13)  and  (14),  the  first  set  of  multipliers 
being  the  natural  sines  of  twice  the  azimuth  of  the  ship's 
head,  the  second  set  being  the  natural  cosines  of  twice  the 
same  azimuths. 

In  these  multiplications  careful  attention  must  be  given  to 
the  signs. 

The  algebraic  sum  of  the  products  in  each  of  the  columns 
(13)  and  (14)  divided  by  3,  f ,  or  unity,  according  as  the 
observations  were  taken  on  24,  the  12  or  8  principal  compass 
rhumbs,  will  give  from  column  (13)  the  approximate  coeffi- 
cient D  and  from  column  (14)  the  approximate  coefficient  E.' 

An  approximate  check  on  D  is  the  mean  of  the  deviations 
on  the  quadrantal  points,  the  signs  of  the  deviations  on  SE. 
and  NW.  being  changed  before  the  mean  is  taken. 

An  approximate  check  on  E.  is  the  mean  of  the  deviations 
on  N.,  S.,  E.,  and  W.,  the  signs  of  the  latter  two  being 
changed. 


140 


NAVIGATION — COMPASS  DEVIATION 


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ANALYSIS  OF  DEVIATIONS 


141 


142  NAVIGATION — COMPASS  DEVIATION 

TABLE  III.— COMPUTATION  OF  EXACT  COEFFICIENTS. 

31,  S3,  <5,  2),  <g. 

ABODE 

Angles  — 0°  31'. 7 +  12°  21'. 5 -5°  05'  +  2°  62'  -  0°  13' 
Sines  —  .0092  +  .2140  -  .0886  +  .05001  —  .00378 
Versines  *  *  *  +  .0232  .00393  .00126  *  *  * 

SC  =  sin  A  =  — .  0092  =  (— ) .  0092 

93  =  sin  B  [1  +  £  sin  D  +  TV  versin  B—  i  versin  C]  +  J  sin  C  sin  E 

= +  .214  [1  +  . 0250 +.001 93  — .00098]  +.0443 X. 00378  =   +.2191 

6  =  sin  C  [1  —  }  sin  D  +  TV  versin  C  —  i  versin  B]  +  1  sin  B  sin  E 

=  — .  0886  [1  —  .025  +  ..0003  - .  0058]  - .  107  X  .00378  =  (— ) .  0863 

£>  =  sin  D  [1  +J  versin  D]  =  +  .  05001  [1  +  .00042]  =  +   .  0500 

6  =  sin  E  —  sin  A  sin  D  =  -.00378  +.0092  X  -05001  =  (-)  .0033 

<£  _  —  .0863  log.         8  . 93601 


tan  x  =  -„-  = 


+  .2197  log.         9.34183 


X  =  21°  26'44"       log.  tan.  9 . 59418 
a  =  338°  3316". 

SECTION  V. 

90.  Determination  of  A. — This  coefficient  is  the  one  which 
expresses  the  proportion  of  the  mean  horizontal  force  north- 
ward of  the  earth  and  ship  to  the  horizontal  force  on  shore 
and  may  be  found  from  equation  (27)  written  as  follows, 

jj'  cos  8 

'  IT  1  +  $  cos  z  —  (£  sin  z  +  ®  cos  2z  —  @  sin  2z  ^61^ 
when  the  horizontal  force  and  the  deviation  for  the  magnetic 
azimuth  z  are  known  in  addition  to  the  exact  coefficient  23, 
(£,  3),  and  @. 

In  the  above  equation 

H'  =  the  horizontal  force  of  earth  and  ship  combined ; 

H  =  the  horizontal  force  of  earth,  considered  as  unity ; 

77'       y7* 

-—  =  -^73-,  T  being  the  time  of  n  vibrations  (say  10)  of  a 

small  horizontal  needle,  3  to  4  inches  long,  on  shore  in  a 
place  free  from  local  magnetic  disturbances;  T'  that  of  the 
same  number  of  vibrations  of  the  same  needle,  the  center  of 
which  is  in  the  same  place  exactly  as  that  occupied  by  the 
center  of  the  compass  needle,  when  the  compass  is  in  place. 


OBSERVATIONS  FOR  HORIZONTAL  FORCE  143 

Great  care  should  be  exercised  in  taking  the  vibrations,  and 
the  mean  of  a  number  of  determinations  should  be  used,  since 
the  error  of  a  single  set  might  be  comparatively  large. 

77'       T2 

The  equation  ~  =  -™  is  true  only  for   infinitesimal  arcs 

of  vibration,  but  may  be  taken  as  sufficiently  exact  for  all 
practical  purposes  if  the  arcs  do  not  exceed  20°.  However, 
the  amplitude  of  arc  should  be  as  small  as  possible  consistent 
with  obtaining  10  well-defined  vibrations. 

The  place  on  shore  where  the  needle  is  vibrated  should  be 
free  from  local  attraction,  a  fact  that  may  be  determined  in 
the  following  way,  namely:  place  a  compass  on  its  tripod 
and  set  up  a  staff  about  50  yards  distant,  note  the  bearing  of 
the  staff  per  compass ;  interchange  tripod  and  staff  and  again 
note  the  bearing  of  staff.  Do  the  same  thing  on  a  line  per- 
pendicular to  the  first  line.  If  the  bearing  and  reverse  bear- 
ing in  each  case  differ  by  180°,  the  locality  may  be  assumed 
free  from  magnetic  local  influences. 

The  horizontal  force  instrument. — This  instrument  is  used 
in  finding  the  ratio  of  the  horizontal  force  on  board  ship  in 
the  position  of  the  compass  to  that  on  shore.  It  consists  of 
a  cylindrical  brass  case,  with  a  removable  glass  cover,  mounted 
upon  a  rectangular  base  which  is  provided  with  levels  and 
leveling  screws. 

The  case  contains  a  horizontal  circle  graduated  to  degrees, 
and  in  the  center  a  pivot  which  supports  a  small  lozenge- 
shaped  magnetic  needle  fitted  with  an  adjustable  sliding 
weight  to  counteract  the  dip  and  capable  of  vibrating  freely 
in  the  horizontal  plane. 

Observations  for  horizontal  force  ashore. — Find  a  level  spot 
free  from  local  attraction,  level  the  horizontal  force  instru- 
ment and  orientate  it.  By  means  of  a  small  magnet  draw  the 
needle  aside  about  20°,  quickly  removing  the  magnet  to  a 
proper  distance.  Then  as  the  needle  passes  the  zero  line  the 


144  NAVIGATION — COMPASS  DEVIATION 

first  time  "  mark  the  time  "  or  start  the  stop  watch,  as  the 
needle  passes  the  zero  line  the  second  time  count  "  one/'  at 
the  next  passage  "two/'  and  so  on  till  the  count  "ten/' 
when  the  time  is  again  noted  or  the  stop  watch  stopped.  The 
interval  of  time  will  be  the  time  required  by  the  needle  to 
make  10  vibrations. 

Observations  on  board. — Observations  are  similarly  taken 
on  board,  the  center  of  the  horizontal  force  needle  occupying 
the  exact  place  usually  occupied  by  the  center  of  the  compass, 
which  with  all  correctors  must  be  removed  to  a  safe  dis- 
tance. The  horizontal  force  instrument  is  leveled  on  a  brass 
table  in  the  compass  chamber,  the  spindle  of  the  table  entering 
the  central  vertical  tube  of  the  binnacle. 

The  magnetic  azimuth  and  deviation  may  be  determined 
by  any  one  of  the  usual  methods. 

The  coefficients  should  have  been  determined  as  accurately 
as  possible  on  equidistant  compass  courses ;  24,  12,  or  8  equi- 
distant headings. 

If  the  observations  be  taken  on  four  equidistant  magnetic 
azimuths  we  will  have 

A  =  iS^cos8  (62) 

because,  the  summation  of  the  sines  and  cosines  being  zero, 
the  exact  coefficients  will  disappear. 


NOTE:  In  the  service  compass,  the  plane  of  the  needles  is  three- 
fourths  of  an  inch  below  the  bottom  of  the  wyes  in  which  the  compass 
rests  when  placed  in  the  binnacle,  and  it  may  be  located  by  placing  in 
the  wyes  a  straight  edge  at  the  center  of  which  is  pasted  a  piece  of 
paper  projecting  vertically  downward  %  of  an  inch. 


COMPUTATION  OF  A 


145 


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146  NAVIGATION — COMPASS  DEVIATION 

Ex.  10. — It  is  required  to  find  A,  from  the  following  observa- 
tions for  horizontal  force  made  ashore  and  on  board  a  moni- 
tor in  the  position  of  the  standard  compass,  the  magnetic 
courses  and  deviations  being  found  by  interpolation  in  the 
Standard's  table  of  deviations,  Art.  55. 

Magnetic  heading.    Deviations.  Horizontal  vibrations. 

North  —    4°  35'  T  14S.60           T  15.66 

East  +  12  00                  16.28 

Sputh  +    4  54                  18.17 

West  —14  09                  17.80 

For  Head,  North.  For  Head,  East. 

15S.66 log  1.19479         15S.66 log  1.19479 

14  .60 log  1.16435         16  .28 log  1.21165 


0.06088  9.96628 

8±  +  4°  35' cos  9.99861    82  +  12°  00' cos  9.99040 

^'cos  84  1.1468,  log  0.05949  $  cos  8,  .9051,  log  9.95668 

a.  n. 

For  Head,  South.  For  Head,  West. 

153.66 log  1.19479          15S.66.  ., log  1.19479 

18  .17 log  1.25935          17  .80 log  1.25042 


9.87088  9.88874 

54'  ____  cos  9.99841       .   84  —  14°  09'.  .  .cos  9.98662 


S*  cos  83  .7401,  log  9.86929         jf  cos  84=.7505,  log  9.87536 
H  •" 

1.1468  +  .9051  +  .7401  +  .7505 
A,=  -  —  —  -  =  .0000 


DETERMINATION  OF  ^  147 

91.  Determination  of  a,  e,  ~b,  and  d,  given  51,  (£,  3),  and  A,. 

.Zfo.  .n.  —  Given  the  following  coefficients  (Exs.  8  and  10), 
9t  =  —  .0092,  @  =  —  .0033,  ©  =  .0500,  A  =  .8856,  it  is 
required  to  find  af  e,  6,  and  d.  See  Art.  79  and  Art.  80. 

a  =  /l(l  +  3))-l  =  .8856  x  1.05  -  1.  =  .9299  —  1.  =  (-).0701 
«  =  A  (1  -  5))  -  1  =  .8856  x  .95  —  1  =  .8413  -  1  =  (-J.1587 

d  —  b  =  2291  =  —  2  x  .8856  x  .0092  =  —  .0163 

d  +  6  =  2A(£  =  —  2  x  .8856  x  .0033=  —  .0058 


2d=  -  .0221  ...  d  =  -.0110 

26=  +  .0105.-.  &  =  +  .0052 


92.  Determination  of  parameters  g  and  ft  and  the  vertical 
force  of  the  earth  and  ship.  —  In  equation 


the  vertical  force  of  the  earth  and  ship  is  expressed  in  terms 
of  the  earth's  vertical  force  as  a  unit  of  measurement. 

7' 

The  mean  value  of  -~  on  two  or  more  equidistant  azimuths 

(J?  \ 
1  -|-  Ic  -f-  —„-  }  of  the  second  member. 

Z' 

Letting  p  be  the  mean  value  of  ^,  or  the  mean  force  down- 

ward of  the  earth  and  ship  in  terms  of  the  earth's  vertical 
force  as  unit,  then 

^=1  +  ^+  f  ;  therefore, 


From  (63),  the  value  of  /*,    -9-  —  ,   and  -r  -    are  derived 

tantf'  tan0 

from  observations  on  4,  8,  12,  or  24  equidistant  courses,  using 
similar  tabular  forms  to  those  used  in  finding  the  approximate 
coefficients. 


148  NAVIGATION  —  COMPASS  DEVIATION 

As  with  the  horizontal  force  instrument,  the  times  of,  n 
vibrations  of  a  dipping  needle  may  be  observed  on  board  and 
on  shore,  the  vibrations  being  made  in  a  plane  perpendicular 
to  the  compass  meridian  on  board  and  perpendicular  to  the 
magnetic  meridian  ashore.  The  dipping  needle  is  correctly 
placed  for  vibrations  when,  at  rest,  the  magnetic  axis  of  the 
needle  is  vertical.  If  T  be  the  time  of  say  10  vibrations  of 
this  needle  on  shore,  and  T'  that  of  the  same  number  of  vibra- 
tions on  board,  the  center  of  the  needle  being  in  the  exact 
position  occupied  by  the  center  of  the  compass  needle,  then 

>7t  /7T2 

*  =  ±—  .     The  times  of  vibrations  are  obtained  in  the  same 

way  as  with  a  horizontal  needle,  the  needle  being  deflected 
from  the  zero  point  about  10°  in  this  case.  The  magnetic  azi- 
muth of  the  ship  may  be  obtained  at  the  same  time  in  any  one 
of  the  usual  ways. 

Owing  to  the  fact  that  the  sine  and  cosine  of  angles  differing 

Z* 

180°  have  opposite  signs,  //,  will  be  the  mean  of  -=•  observed  on 

two  opposite  magnetic  headings. 

Eegarding  h  as  zero,  if  g  is  known,  we  may  find  ^  from  one 

Zf 

observation  of-^-;    if  one  observation  is  made  on  magnetic 

£j 

Z' 

East  or  West,  g  will  disappear  and  /*,  will  equal  -—  . 

Z' 

If  observations  for  -^-  are  made  on  four  equidistant  mag- 
£ 

netic  headings,  then 


fjL=lS..  (64) 

If  the  observations  are  made  on  N".,  E.,  S.,  and  W.,  mag- 
netic, then  JJL  may  be  found  from  all  four,  g  from  the  observa- 
tions at  N".  and  S.,  li  from  those  at  E.  and  W.,  a  fact  that  is 
apparent  from  a  consideration  of  equation  (63). 


DETERMINATION  OF 


149 


Ex.  12. — It  is  required  to  find  //,  g,  and  h  from  the  following  observa- 
tions for  vertical  force  made  ashore  and  on  board  a  "  Monitor  "  in  the 
position  of  the  standard  compass,  tan  6  being  2.86. 


Mag.  heading. 

North 
East 
South 
West 

For  Head,  North. 

19S.75 log  1.29557 

18.43 log  1.26553 


-p=  1.1484...  log  0.06008 

/v 

For  Head,  South. 

19S.75 log  1.29557 

19.08 log  1.28058 

0.01499 

2 


Vertical  vibrations. 

T'  18S.43  T  19S.75 

18.94 
19.08 
19.00 

For  Head,  East. 

19S.75 log  1.29557 

18.94 log  1.27738 

0.03004  0.01819 

2  2 

Z£  =  1.0874...  log  0.03638 

For  Head,  West. 

19S.75 log  1.29557 

19.00 log  1.27875 

0.01682 

2 


~?  =  1.0715.  .  .log  0.02998   ^  =  1.0805.  .  .log  0.03364 
+  1.0874  +  1.0715  +  1.0805 


4 

On  K,  —^ 

=  1.1484 

On  West,  ~ 

On  S.,^f 

=  1.0715 

On  East,  : 

2)0.0769 

2)  —  0.0069 

-£-=  0.0384          5-^-27  =  —  °-0034 
tan  6  tan  0 

g  —  .0384  X  2.86  =  .1098    h  —  —  .0034  X  2.86 

=  (— )  .0097 

9  .  being  the  value  of  —^  at  N".  —  the  value  of  -^  at  S. 
tan  0  Z  Z 

^h    being  the  value  of  -'  at  W.  —  the  value  of  ^L  at  E. 
tan  e  Z  Z 


150  NAVIGATION-  —  COMPASS  DEVIATION 

SECTION  VI. 

93.  Other  methods  of  finding  the  exact  coefficients  93,  (£, 

and  2).  —  In  the  case  of  a  compass  well  located  on  board  ship, 
3t  and  (£  are  either  zero,  or  very  small,  and  for  all  practical 
purposes  may  be  neglected  without  appreciable  error.  The 
equations  for  deviations  then  become  very  simple  on  the  two 
cardinal  and  the  intercardinal  points  of  any  quadrant.  From 
observations  made  on  such  points  the  compass  may  be  fairly 
well  compensated,  and,  in  the  case  of  one  already  compen- 
sated, a  very  good  residual  curve  may  be  obtained  by  sub- 
stituting the  resulting  values  of  93,  (£,  and  S)  in  equation  (34), 
as  illustrated  in  Art.  97.  This  is  a  good  method  whether  at 
sea  or  at  anchor  in  port  swinging  to  tide.  Even  if  at  a  dock, 
a  vessel's  head  may  be  sprung  around  sufficiently  to  get  the 
required  observations. 

When  practicable,  choose  that  quadrant  in  which  the  direc- 
tive force  on  the  needle  is  strong. 

In  NE.  quadrant.  —  Letting  51  and  @?  be  zero,  we  shall  have 
for  compass  courses  North,  NE.,  and  East,  from  (30),  the  fol- 
lowing : 

(1)  sin  80  =  <£  +  ®  sin  80  .'.  <£  =  sin  80(1  —  3)). 

(2)  sin  8,  =  33£3  +  <5£8  +  3)  cos  88. 

(3)  sin  S6  =  $  —  ®  sin  S6  .'.  93  =  sin  86(1  +  ®). 
Substituting  value  of  33  and  (£  in  (2), 

sin  83  =  £3[sin  80(1  —  ©)+  sin  8C(1  +  $)]+  ®  cos  83. 


+  ©  -j  cos  83  —  Si  (sin  80  —  sin  86)  }• 
sin  88  — 


cos  83  —  #3  (sin  80  —  sin  86) 


OBSERVATIONS  IN  ONE  QUADRANT  151 

Multiplying  through  by  $3,  and  as  #32  =  -J,  we  have  for  the 
NE.  Quadrant, 


_  £3  sin  8,  —  •&  (sin  80  +  sin  86) 
S3  cos  83  —  i(sin  80  —  sin  86) 
23=(l  +  S))sin86 
<£=(!  —  £>)sin80 


(£  =  —  (1  —  2))sin812 

And  for  the  SW.  Quadrant, 

#3  sin  815  —  |  ( sin  812  -f-  sin 


'18 


(65) 


Similarly  for  the  SE.  Quadrant, 

-  83  sin  89  +  £  (sin  86  +  sin  812) 

— i(sin86  — sin812)    ,  ,„, 


?8cos81B-— 4(sin812  —  sin818)  L  ^ 


And  for  the  NW.  Quadrant, 

_    —  83  sin  821  -\-  %  (sin  818  -\-  sin  80) 

#3  cos  821  —  i(sin  818  —  sin  80)  ,  (     . 

93  =  —  (l  +  3))sin818 
£  =      (1  — S))sin80 

If  observations  have  been  made  on  the  three  cardinal  points 
and  two  intercardinal  points  of  one  semicircle,  consider  each 
quadrant  of  that  semicircle  separately,  find  the  values  of  58,  (£, 
and  3),  from  each,  and  take  the  mean  of  the  two  determinations 
of  each  coefficient;  or,  combine  the  formulae  in  the  proper 
quadrants  before  proceeding  with  the  computation. 


152  NAVIGATION — COMPASS  DEVIATION 

Ex.  13. — A  distant  object,  the  magnetic  bearing  of  which  was 
326°  45'  bore  (p.  s.  c.)  respectively  355°,  335°,  and  328°,  as 
the  ship  headed  (p.  s.  c.)  successively  South,  SW.,  and  West. 
Eequired  33,  (£,  and  3). 
Deviation  on  South  =  812  =  —  28°  15', 

sin  812  =  —  .473,  cos  812  —  +  .881. 
Deviation  on  SW.  =  S15  =  —  8°  15', 

sin  815  =  —  .143,  cos  815  =  +  .990. 
Deviation  on  West  =  S18  =  —  1°  15',  sin  818  =  —  .022. 


_  .707  x  (-.143)  _  i  (-.473  -  .022) 
~  .707  x  .990  —  \  \  —.473  —  (—.022)  [- 

-.1011 +.2475       .1464 


+  .1582 


.6999  +.2255       .9254 
5)3  _  _  (1  +  .1582)(—  .022)  _      1,  1582  x.  022=  +.0255 

(£  =  —  (1  —  .1582)(—  .473)=  +.8418  x  .473  =  +.3982 

94.  Determination  of  33  and  (£  from  observations  of  devia- 
tion and  horizontal  force  on  one  heading.  —  Regarding  51 
and  @?  as  zero,  that  is  &  and  d  as  zero,  we  have  from  (20)  and 


cos  z'  =  (1  +a)  cos  z  +  c  tan  0  + 

or,  jp  cos  «'=(!  +  a)  cos  z  +  A$.  (69) 

—  |^sin  z'  =  —  (1  +  e)  sin  z  +  /  tan  0  +  Q. 
H  n. 

or  —  ^sin  z'  =  —  (1  +  e)  sin  z  +  AS.  (70) 

H 

Substituting  the  values  of  (1  +  a)  and  (I  -\-  e)  from  (35) 
and  (36),  transposing  and  dividing  through  by  A,  we  have, 

33  =  -^cos  zf  —  (1  +  2))  cos  -z.  (69a) 

A/2 

S  =  —  -^sin  z'  +  (1  —  £>)  sin  a.  (70a) 

x/z 

2'  is  the  azimuth  of  the  ship's  head  per  compass;  z  is  the 
magnetic  azimuth  of  the  ship's  head  and  may  be  determined  by 


OBSERVATIONS  ON  Two  HEADINGS  153 

a  time  azimuth  of  the  sun,  from  the  bearing  of  a  distant  object 
of  known  magnetic  bearing,  or  by  reciprocal  bearings ;  after  the 
above  data  have  been  obtained,  remove  the  compass  to  a  suffi- 
cient distance  and  take  vibrations  of  a  horizontal  needle  in 
the  exact  place  of  the  compass  needle,  calling  T'  the  time  of 
10  vibrations.  Take  vibrations  of  the  same  needle  ashore, 

H'      T2 

calling  T  the  time  of  ten  vibrations  there,  then  -77-=  777,2. 

£1  J- 

Therefore,  if  A  and  ®  can  be  obtained;  23,  (£,  and  a  may 
be  found. 

These  coefficients,  A  and  £),  are  so  nearly  the  same  for  com- 
passes in  similar  positions  in  similar  ships  that,  in  the  absence 
of  any  better  values,  they  may  be  taken  as  the  same  as  those 
in  a  sister  ship,  or  assumed. 

With  the  approximate  values  of  23  and  (£  and  the  assumed 
value  of  3D,  the  compass  may  be  roughly  corrected  when  in  dry 
dock,  moored  to  a  wharf,  or  when  it  is  impossible  to  get  ob- 
servations on  more  than  one  heading,  provided,  however,  that 
there  is  no  other  iron  vessel,  nor  other  causes  of  disturbance, 
sufficiently  near  to  exercise  magnetic  influence.  If  the  com- 
pass should  not  be  compensated,  then  a  table  of  approximate 
deviations  may  be  made  by  the  formulae  of  Art.  97. 

Such  observations  may  be  valuable  in  determining  the  loca- 
tion of  compasses  for  ships  while  still  on  the  stocks. 

95.  Determination  of  23,  (£,  3),  A,  a,  and  e  from  observations 
of  deviation  and  horizontal  force  on  two  headings,  51  and  (£ 
heing  neglected. 

Let  z\  and  z'2  be  the  two  compass  headings;  z^  and  z2 

TT'  fTJ2 

the  two  magnetic  headings ;  -==*  =  -7=>r  the  horizontal  force  of 

H         -/i 

TT'  rpi 

earth  and  ship  on  the  first  heading  in  terms  of  H;  -™i  —  7^ 

JJ.  -L  a 


the  same  on  the  second  heading.     Then  we  have  from  (69) 
and  (70)  for  the  two  headings : 


154 


NAVIGATION — COMPASS  DEVIATION 


=  £±  cos  z(  —(l  +  a}  cos  zl        (71) 
jti 


=  Jj*  cos  z2'  — 

•»       jfZ 


01   (1  +  fl)  = 


cos  za        (72) 


H' 
cos  Zi  —  ^=2cos  zi 


TT' 

n^  si 


f  (COS  Zi  —  COS  Z2) 

(1  +  e)  sin  zx        (74) 
(1  +  e)  sin  z2        (75) 


or 


i  (sin  zl  —  sin  z2) 
From  (73)  and  (76),  (35)  and  (36), 

A  =  i{(l  -f-^)  +  (1  +  e)} 


Adding  (71)  and  (72)  and  dividing  by  A, 

1  f  1  (H[         ,       H^         \ 

%  =  +  7-  1  TJ-  ^  ^  cos  zt  +  -^-cos  z2  1 

—  (1  +  0)  [_-^-(cos  Zi  +  cos  z2)J  I 
Adding  (74)  and  (75)  and  dividing  by  A? 

i  f  i  /#  ,    .  ,  ^r;  .    A 

-T\T^811]^+^sm^ 

-  (1  +  e)  r^-  (sin  zi  +  sin  z2)  J  | 

(£ 
tan  a  =    -' 


(73) 


(76) 

(77) 
(78) 


(79) 


(80) 
(81) 


Since  the  sines  and  cosines  of  angles  differing  180°  have 
opposite   signs,  if  the  observations   are  taken   on   magnetic 


OBSERVATIONS  ON  Two  HEADINGS  155 

courses  diametrically  opposite,  (1  +  a)  in  (79)  and  (1  +  e) 
in  (80)  will  disappear,  J(cos  z±  —  cos  zz)  will  become  cos  % 
and  -J  (sin  z^  —  sin  zz)  will  become  sin  zv ;  and  equations  (73), 
(76),  (79),  and  (80),  being  much  simplified,  will  become 

If    T-T'  TT'  ~\ 

1   **i            i         -"i  t  (  /oo^ 

1.   +   a  =  -£  <  -rfCQS  Z^  -TT   COS  Z2   f  V^*^ 

COS  Zt 

1  +  e  =  o-(  -^-sin  z[  —  -rr  sin  «;  }  (83) 


=  T  {  T  ( 


sin  Zx 
^OOS^+^COB*;)]-  (84) 


(85) 


This  method  is  strongly  recommended,  and  most  excellent 
results  may  be  obtained  when  the  vibrations  are  carefully 
taken. 

In  the  steering  compasses  of  some  of  the  battleships,  the 
ship's  force  exceeds  that  of  the  earth  and  it  is  impossible  to 
obtain  a  curve  of  deviations  for  such  compasses  by  swinging 
ship,  and  resort  must  be  had  to  vibrations. 

However,  care  must  be  exercised  in  selecting  the  rhumbs,  as 
the  formulae  will  fail  if  the  magnetic  azimuths  are  equally 
distant  from  any  one  of  the  cardinal  points;  for  if  equally 

distant  from  North  or  South,  (1  +  a)  reduces  to  the  form  —  ; 

and  if  equally  distant  from  East  or  West,  (1  +  e)  takes  that 
form.  As  a  general  rule,  select  rhumbs  on  or  near  opposite 
quadrantal  points,  or  on  or  near  two  adjacent  cardinal  points. 
This  method  is  valuable  in  locating  compasses  on  board  new 
ships  and  in  determining  beforehand  the  forces  of  the  ship, 
and,  if  desired,  a  deviation  table.  Select  the  different  places 


156  NAVIGATION — COMPASS  DEVIATION 

where,  for  other  reasons,  the  compass  might  be  located ;  obtain 
the  compass  and  magnetic  headings  and  the  time  of  10  vibra- 
tions of  the  horizontal  needle  in  the  exact  position  to  be  occu- 
pied by  the  compass  needle  when  the  ship  is  on  the  first 
heading;  change  the  position  of  the  ship  and  do  the  same  on 
the  second  heading;  note  the  time  of  10  vibrations  of  the 
same  horizontal  needle  ashore.  Proceed  with  the  computa- 
tion, and  all  other  considerations  being  equal,  select  that  spot 
as  the  best  location  for  the  compass,  where  A  is  greatest. 

The  following  form  not  only  facilitates  the  solution  but 
indicates  the  data. 


OBSERVATIONS  ON  Two  HEADINGS 


157 


rH  rH 
t-  Z> 

rH 

t* 

N 

O  o 

^  o 

a 

t-  t- 

s° 

. 

1  + 

T 

I>  J> 

^ 

N 

o  o 

o 

« 

o 

t-  1, 

CO  O 

B1^ 

o 

I  + 

T 

< 

Si 

o 

t» 

O  CM 

w 

^_^  rH 

g 

CO  OS 

s-^    TH 

'53 

^ 

^_*  ^ 

••-^  ^ 

• 

^N 

i  + 

+ 

*L 

« 

% 

O> 

o 

TH    0? 
«0    rH 
00   I> 

t- 

^  o 

„§ 

•<* 

T- 

cc 

b- 

•"±1           *~ 

0 

^^   .  * 

^.^^      * 

>—  '  ^—  ^ 

?. 

Si  J*i 

_  + 

I 

J, 

o 
^       1 

.5 

00  •* 
CO  JO 
CO   OS 
CO  t~ 

a 

CQ 

50 

5 

1 

v^        0 

II    S 

«    M. 

CQ 

^  + 

Hn 

HM 

rji             -s 

og     1 

CO   CO 
<M   CO 

T05'     1 

M 

•*  O 

J,     J, 

09 

9               . 

1H         ^ 

0 

•-» 

S       II      5 

O 

i  + 

0<                      "                    s., 

*                          ^-N                          ^ 

CO 

l^         O 

«  co      +        22.        4- 

Eifi 

XO 
00 

CO  ^-1 

CO 

TH 

TH 

CM          «O       c, 

J>          00       C 

co       os     c 

§  s  §    II     «    d 

5      «    ^   co  u       4- 

ii 

TH     t- 

O5   TH 

TH 

0             TH         C 

5  °°-i   -    i 

rtj     [^ 

d  TH 

+ 

+ 

^.  1-   r    2 

II 

II     II    1 

1    II  II    II     p     f 

ft, 

CO*  •*' 
CO  CO 

III 

ig  a  a 

5    Z>      CQ  CQ 

;  x 

«g'§|b^w 

Q     ft     «  r     rH  |r<     rH  |« 

1         "*     ii         II         II 

bo 

T~  ^ 

f 

&      ft       te 

CO  CO 

n" 

V 

0 

OS  <M 
OS  O 

TH 

o"  t-  «c 

II 

>"  00s 

II 

00 
0  0 

+ 

+ 

ti 

o 

T^ 

T-J 

03 

03    ti 

<o  a 

TH   CM 

w 

158 


NAVIGATION — COMPASS  DEVIATION 


W 


s'l 


ii' 


60   A 
P     O 


A 

.,_,    03 

03     CO 


•    I 


02  -~ 

O  g 

55  O 
i— I 

Q  02 

<t  . 

H  p 

W  o3 

§1 


|8 


+  e 


38 


+ 


II 


+ 


$M 

oo 


COCO 


-f 


^  Vwfi^SSS 

@e  « 


s  x 

"H  JP 

Jl      II 


THE  Two  PARTS  OF  23  AND  (£  159 

96.  Determination  of  the  forces  of  hard  and  soft  iron  caus- 
ing semicircular  deviation. — We  have  seen  that  semicircular 
deviation  is  caused  by  two  separate  forces :  (1)  the  force  due 
to  subpermanent  magnetism,  the  components  of  which  to  head 
and  to  starboard  in  terms  of  the  "  mean  force  to  North  "  as 

P          0 

unit  are  respectively  ™-and  ^  ;  (2)  the  force  due  to  tran- 
sient magnetism  induced  in  vertical  soft  iron,  the  components 
of  which  in  the  same  axes  in  terms  of  "  mean  force  to  North  " 

as  unit  are  respectively  -y  tan  0  and  L-  tan  6. 

In  the  above  expressions  for  the  forces,  the  values  -T-,  -p  -j 

and  •£-  are  the  parts  that  do  not  change,  and  it  is  desirable 

to  determine  them  at  the  first  opportunity,  and  then  the  forces 
due  to  hard  and  soft  iron  separately  at  the  place  selected  as 
that  of  compensation. 

These  quantities  may  be  determined  from  two  different 
values  of  $8  and  &  observed  at  places,  in  widely  different  lati- 
tudes, where  the  dip  and  horizontal  force  are  known.  When 
they  are  once  determined,  it  will  be  possible  to  correct  sepa- 
rately the  deviation  due  to  the  two  kinds  of  iron,  that  due  to 
hard  iron  being  corrected  by  magnets  and  that  due  to  soft  iron 
by  what  is  known  as  a  Flinders  bar.  This  bar  is  of  soft  iron 
and  is  about  36  inches  long.  It  is  placed  in  the  starboard 
angle  (av)  of  the  forces  due  to  induction  in  c  and  /  with  the 
lower  end  level  with  the  compass  needles;  or  in  the  angle 
180°  +  av,  the  upper  end  in  this  case  level  with  the  compass 
needles.  When  /  is  zero,  av  is  0°  or  180°. 

The  Flinders  corrector  consists  preferably  of  a  bundle  of 
rods  of  about  J  inch  diameter  and  about  36  inches  in  length, 
so  that,  at  a  fixed  distance,  the  intensity  of  its  induced  force 
may  be  varied  as  desired  by  increasing  or  decreasing  the  num- 
ber of  the  rods,  instead  of  varying  the  distance  of  a  single  rod. 


160 


NAVIGATION — COMPASS  DEVIATION 


If  whilst  a  vessel  is  on  the  magnetic  equator,  the  compass 
be  carefully  compensated,  the  force  causing  semicircular  de- 
viation, being  there  due  to  subpermanent  magnetism  alone, 
should  be  entirely  neutralized  by  magnets.  Then,  if  on  change 
of  magnetic  latitude,  a  Flinders  bar  be  so  placed  as  to  correct 
any  semicircular  deviation  that  appears,  the  compensation 

should  be  general  for  existing  conditions. 

p 
If  the  compass  is  not  compensated,  with  the  values  of  -p 

c     0  f 

•p  -j-,  and  jj-  known,  it  will  be  possible  to  predict  changes  in 

the  deviations  and  to  make  out  a  table  of  deviations  for  the 
locality  of  other  cruising  grounds,  H  and  6  being  known. 

Letting  the  .values  of  the  quantities  proper  to  the  problem 
be,  at  the  first  place  231?  (£±,  H19  tan  01?  and  at  the  second  place 
932  ,  (£2  ,  H2  ,  tan  02  ,  we  shall  have 


=  -•  +  —  tan 


JETJB,  = 


P^ 

X 


~  tan  02#2 


=  -£  +  -4-  tan 


(86) 


##.  m— At  Cape  Henry  $B±  =  +  .697,  ^  =  —  .119, 
H;  =  .219,  tan  0±  =  2.605;  at  Key  West  932  =  +  .393, 
(£2  =  —  .08,  H2  =  .311,  tan  02  =  1.393.  It  is  required  to 
find 

P      c       Q          ,    f 
T>  "7  '    l  >  and  T* 

A  A  A  A 

(2)  What  were  the  values  of  23  and  (£  at  New  York, 
£  =  .185,  tan  0=3.14? 

(3)  What  was  the  total  force  at  New  York  due  to  subper- 
manent  magnetism,  and  what  was  the  total  force  due  to  tran- 
sient magnetism  induced  in  vertical  soft  iron  ? 


THE  Two  PARTS  OF  23  AND 


161 


(4)  A  Flinders  bar  having  been  placed  while  the  ship  was 
at  New  York,  it  is  required  to  find  the  direction  in  which  it 
was  placed,  and  the  compass  heading  after  correction,  if  be- 
fore, the  vessel  headed  48°  30'  (p.  c.),  dev.  ~H  41°  30',  the 
azimuth  of  the  ship's  head  remaining  the  same. 

By  substitution  in  (86)  we  have 


.219  X  .697  =  4-  +  T-  X  2.605  X  .219 


.153  = 


.128 


-       +  .433-- 


.031  = 


.137  -f- 


X  .393  =  ~  +  •£-  x  1.393  X  .311 


.122=4  + 
-£  =  .024 


.219  X  (-  .119)  =  -f-  +  ^  X  2.605  X  .219   .311  X  (—  .08)  =  -J-  +•  ^-  X  1.393  X  .311 
(-)  .026  =  -§-  +  .570  -f  (-)  .025  =  -^  +  .433  £ 


(-)  .025  = 


(-)  .001  = 


.137 


4- 


(-).025  =  -5 .003 

-AL=(-).022 


-f=(-)  .007 


(2)   To  find  the  value  of  93  and  (£  at  New  York, 
P         c   , 


+  ((-)  -007  X  3.14) 


=  y     +  -226  X  3.14       (S"" 

=  .130  +  .710  =  +  .840      (£  =  (—  )  .119  —  .022  =  (—  )  .141 

(3)   +  .130  =  force  in  keel  line  due  to  subpermanent  mag- 

netism at  New  York. 
(  —  ).119  =  force  transverse  to  keel  line  due  to  subper- 

manent magnetism  at  New  York. 
+  .710  =  force  in  keel  line  due  to  vertical  soft  iron  at 

New  York. 

(  —  )  .022  =  transverse  force  due  to  vertical  soft  iron  at 
New  York. 


162  NAVIGATION — COMPASS  DEVIATION 

tana..,  ^••(^^B(-)-JlM 

therefore  as^  =  317°  31'  45" 
tan  a,  =  ^=y|^  =  (-)  .03099 

therefore  av  =  358°  14'  +  . 
Total  force  due  to  subpermanent  magnetism 

—  V(-130)2  +  (— .119)2  =  .176. 
Total  force  due  to  vertical  soft  iron 


+  (—  .022)2  =  .710. 

(4)  The  Flinders  bar  is  placed  in  the  angle  av  =  358°  14'+ 
with  its  lower  end  on  a  level  with  the  compass  needles,  or  in 
the  angle  av  +  180°  =  178°  14'  +  with  its  upper  end  on  a 
level  with  the  compass  needles,  and  at  such  a  distance  as  to 
neutralize  the  deviation  due  to  the  vertical  soft  iron.  In  case 
the  corrector  is  to  be  at  a  fixed  distance  from  the  compass, 
then  increase  or  decrease  the  number  of  rods  till  the  desired 
effect  is  produced. 

The  deviation  resulting  from  the  equation 

—  tan  0  sin  z  +  -*r  tan  0  cos  z 
tan*=  —  -jr-       —       (a) 

1  -f-  -£-  tan  0  cos  z  —  L-  tan  0  sin  z 

A  A 

is  the  deviation  due  to  vertical  soft  iron  on  the  magnetic  head- 
ing 2. 

In  the  example  zr  =  48°  30',  dev.  =  +  41°  30',  and  there- 
fore z  is  East. 

Substituting  the  values  of  j  tan  0  and  -^-  tan  0  found  in 

part  (2)  of  this  example,  we  have 

*  _  +  -710  X  1  +  (-  .022)  X  0  .710 

=  1+.710  xO  — (-.022  X  1)  ""  1.022 
=  .6947  .'.  d  =  +  34°  47'. 


DEVIATIONS  FROM  THE  COEFFICIENTS  163 

The  deviation  due  to  vertical  soft  iron  being  +  34°  47'  on 
the  given  heading,  that  amount  should  be  removed  by  the 
Flinders  corrector;  therefore,  after  the  correction  has  been 
made,  the  compass  heading  should  be  83°  17'. 

97.  Computation  of  deviations  from  the  coefficients.  —  Vari- 
ous methods  have  been  explained  for  obtaining  new  values  of 
the  coefficients,  especially  the  changing  ones  ;  having  obtained 
these  it  may  be  desirable  to  compute  a  deviation  table;  or 

knowing  the  values  of  j,  -j,  Q   and  y,  it  may  be  necessary  to 

determine  the  deviations  for  certain  localities  to  be  visited 
where  there  may  be  no  opportunities  for  swinging  ship. 

From  the  approximate  coefficients  the  deviations  may  be 
obtained  from  the  equation 

8  —  A  +  B  sin  z'  +  C  cos  sf  -f  D  sin  2sf  +  E  cos  2z'; 
and  from  the  exact  coefficients,  they  may  be  found  for  mag- 
netic azimuths  from  equation 

%  +  %  sin  g  +  (£cos  s  +  g)  sin  2z  +  (£  cos  2z 
~  I  +  23  cos  z  —  (£  sin  z  +  SD  cos  2z  —  (£  sin  2z' 

then  by  use  of  Napier's  diagram  the  deviations  may  be  found 
for  the  compass  headings. 

However,  deviations  are  desired  for  compass  headings  and 
may  be  found  for  such  from  equation  (34), 


-—  ~  — 
The  following  form  facilitates  the  computation. 


164 


NAVIGATION — COMPASS  DEVIATION 


00 

<B 
ft 

-t-> 

03  o 

1J 

j 

3 

-8558S8S8S 

Ot-COt-t-tr-«Ol-lO 

0 

O  + 

a 

1             1             1             1          +        +        +          1 

0 

(3 

0 

^  d^ 

1  "** 

03 

«2> 

§N 

Pa- 
ll} 

CQ      S 

1      !      1      1     +    +    +     1 

T3 

0) 

,c 
+a 

^ 

! 

illiilli 

T3 
{3 

1-1  5 

J 

£ 

- 

^^, 

eg 

CJ 
OQ 

iilosiil 

2 

O 

+        1        +        1 

V 

N 

~T~ 

v?t 

© 

II 

% 

rH         O         r-t         O         i—  1 

TS 

a 

o3 

§ 

| 

r»T 

s 

|  i  i  §  1  1  §  1 

O 

i—  l 

00 

1    1    1    1    +  +  +    1 

+ 
II 
& 

o 

a 

1  1  1  1  1  1  i  1 

to, 

OQ 

+       1       +       1 

CO 

"i  ' 

o 

II 

^ 

rl 

N 

O        TH        O         i—  i         O 

T 

d 

'S 

1 

K 
Ml 

tf 

i 

"N 
1 

f    l'       +  +  +        1 

w    «3 

T 

to 

^     w 

T| 

II 

(2) 

i 

,-H       OQ       0       CC       --I 

1 

II       0 

o 

II     ° 

8P  £ 
a    ^ 

a 

lilillll 

Bj 

*3 

e 

III        +  +  + 

o  a 

1 

1  S 

t-    *-< 

II 

s? 

a 

0        t^       rH        CC        0 

H.-a 

DO 

1*3    '^ 

•O' 

•x 

a 

03 

o 

oojogjog^o^o 

'C 

P< 

K 

P< 
—  ^ 

~       *       C*        **       CO 

HEELING  ERROR  165 

It  may  be  noted  that  the  second  halves  of  columns  33  sin  z' 
and  &  cos  z'  are  the  same  as  the  upper  halves  with  a  change 
of  signs.  Column  2)  sin  2z'  is  the  same  in  the  lower  half  as 
in  the  upper  half,  no  signs  changed;  the  same  is  true  for  the 
column  ®  cos  2z'. 

SECTION  VII. 
Heeling  Error. 

98.  In  Art.  74  it  was  shown  that,  with  the  ship  upright,  the 
magnetic  forces  acting  on  the  compass  needle  to  head,  to  star- 
board, and  vertically  downward  were  expressed  by  Poisson's 
equations,  in  which  X,  Y,  and  Z  were  the  components  of  the 
earth's  force  in  the  three  directions  called  respectively  the  axes 
of  X,  Y,  and  Z,  the  first  two  being  in  the  horizontal  plane; 
that  magnetism  was  induced  in  the  parameters  o>  6,  c  .  .  .  .  & 
by  the  earth's  component  parallel  to  the  direction  in  which  the 
parameters  lay,  the  induced  force  in  each  case  being  a  linear 
function  of  the  inducing  force. 

When  the  ship  heels,  the  transverse  and  vertical  iron  alter 
their  directions  and  make  with  the  axes  of  Y  and  Z,  respec- 
tively, an  angle  equal  to  the  angle  of  heel ;  for  Poisson's  equa- 
tions to  express,  under  the  new  conditions,  the  forces  to  head, 
in  the  inclined  transverse  direction  of  the  deck,  and  in  the 
inclined  direction  of  the  keel,  these  directions  must  be  taken 
as  new  axes,  and  the  earth's  force  resolved  parallel  to  them. 

Let  the  resolved  components  of  the  earth's  force  be  YI  in  the 
inclined  transverse  direction  of  the  deck,  Zt  in  the  inclined 
direction  of  the  keel,  the  force  X  to  head  being  unchanged  by 
heeling. 

The  force  induced  in  the  fore-and-aft  iron  will  be  the  same 
as  before  the  ship  was  heeled,  the  force  induced  in  the  trans- 
verse iron  will  be  the  same  linear  function  of  Yt  as  it  formerly 
was  of  Y,  and  that  induced  in  the  iron  formerly  vertical  will 


166 


NAVIGATION — COMPASS  DEVIATION 


be  the  same  linear  function  of  Zi  as  it  formerly  was  of  Z, 
because,  the  axes  and  the  iron  being  parallel,  the  ratio  of  the 
earth's  component  in  the  axis  and  the  force  induced  in  the 
iron  of  that  axis  will  not  be  changed  by  heeling,  and  hence  the 
values  of  the  parameters  in  the  equations  for  the  new  axes 
remain  unchanged. 

In  other  words,  the  rods  a,  d,  and  g  will  be  magnetized  by 
force  X;  ~b,  e,  and  Ji  by  force  YI;  c,  /,  and  Jc  by  force  Zt;  whilst 
the  components  of  the  subpermanent  magnetism  remain  un- 
changed. 


FIG.  52. 

Therefore,  letting  X',  Yj,  and  Zi  represent  respectively 
the  forces  of  earth  and  ship  in  the  new  axes  to  head,  to  star- 
board, and  to  keel,  Poisson's  equations  become : 

X  '  =  X  +  aX  +  lYi  +  cZi  +  P.  (87) 

Y;  =  Yi  +  dX  +  eYi  +  fZi  +  Q.  (88) 

Zi  =Zi  +  gX  +  hYi  +  JcZi  +  R.  (89) 

The  next  step  is  to  express  the  forces  represented  by  equa- 
tions (87),  (88),  and  (89),  in  terms  of  the  components  X,  Y, 
and  Z ;  to  do  which  it  will  be  necessary  to  substitute  the  values 
of  Yi  and  Zi  in  terms  of  those  quantities. 


HEELING  ERROR  167 

In  Fig.  52 ,  let  OY  and  OZ  be  the  transverse  and  vertical 
axes,  ship  upright;  OY i  and  OZi  the  corresponding  axes, 
ship  heeled  i°. 

OA  =  Y,  the  horizontal  component  of  earth's  force  to  star- 
board ; 

OB  =  Z,  the  vertical  component  of  earth's  force ;  then 
0 T  is  that  component  of  the  earth's  total  force  that  acts  in  an 
athwartship  plane  through  the  compass,  on  the  North 
point  of  the  needle  at  0,  which  changes  neither  in 
direction  nor  intensity  when  the  ship  heels;    and 
hence 
OR  =  Yi  is  the  component  of  the  earth's  force  in  the  new  axis 

to  starboard,  and 
OL  =  Zi  is    the  component  of  the  earth's  force  in  the  new 

axis  to  keel. 
But  from  the  figure 
OR  =  ON  +  NM  +  MR  =  OA  cos  i 

+  (AM  +  MT)  sin  i  =  OA  cos  t  +  OB  sin  i 

orYi  —  Y  cos  t  +  Z  sin  i,  (90) 

and  OL  =  OK  —  LK  =  OK  —  FB  =  OB  cos  i  —  BT  sin  i 
or  Zi  —  Z  cos  *  —  T  sin  i.  (91) 

Substituting  (90)  and  (91)  in  (87),  (88),  and  (89),  and 
collecting  the  terms  with  common  factors,  we  have 
X'  =  X  +  aX  +  (  I  cos  i  —  c  sin  i)  Y 

+  (b  sin  i  +  c  cos  i)  Z  +  P,  (92) 

Y'i  =  (cos  i  +  e  cos  i  —  /  sin  i)  Y  ~f-  dX 

+  (sin  i  +  e  sin  i  +  /  cos  i)  Z  +  Q,  (93) 

Z'i  =  (cos  i  -{-  h  sm  i  -\-  Je  cos  i)  Z  -\-  gX 

—  (sin  i  —  h  cos  i  +  k  sin  t)  Y  -\-  R,  (94) 

which  are  the  forces  acting  still  in  the  new  axes,  though 
in  terms  of  X,  Y,  and  Z. 

As  the  compass  needle  is  constrained  to  move  in  the  hori- 
zontal plane,  to  obtain  an  expression  for  deviation  due  to  the 


168  NAVIGATION — COMPASS  DEVIATION 

above  forces,  we  must  obtain  their  components  in  the  hori- 
zontal plane. 

Since  the  ship  is  heeled  about  the  axis  of  X,  the  force  X' 
in  equation  (92)  is  already  acting  in  the  horizontal  plane, 
and  it  is  only  necessary  to  obtain  the  horizontal  component 
of  earth  and  ship  to  starboard  represented  by  F'. 

Eef erring  again  to  figure  52, 

Let  OP  represent,  in  intensity  and  direction,  that  component 
of  the  total  force  of  earth  and  ship  which  acts  in  an 
athwartship  plane  through  the  compass  after  the  ship 
has  heeled  i°,  then 

OR"  is  the  component  of  that  force  in  the  new  axis  to  star- 
board, and 
Yf  =  OA"  is  the  component  of  that  force  in  the  horizontal 

plane  to  starboard. 
But  from  the  figure: 
OA."  =  OA'  —  A" A'  =  OA'  —  R"  8 

—OR"  cos  i  —  R"P  sin  i  —  OR"  cos  i  —  OL"  sin  i; 
or  T  -  OA"  —  Y'i  cos  t  —  Z'«  sin  t.  (95) 

Substituting  in   (95)   the  values  of  Y'i  and  Z't  from   (93) 
and  (94),  we  have 

Y'  =  (cos2  i  +  e  cos2  i  —  /  sin  i  cos  i)  Y  +  d  cos  i  X 
+  (sin  i  cos  i  -\-  e  sin  i  cos  i  -f-  /  cos2  i)  Z  -(-  Q  cos  i 

—  (sin  i  cos  i  -\-  h  sin2  i  +  fc  sin  i  cos  i)  Z  —  g  sin  i  X 
-f-  (sin2  i  —  Ti  sin  i  cos  i  -j-  k  sin2  i)  Y  • —  R  sin  i. 

Y*  •=  {  sinH'-f-cos2i  \  Y-}-  \  dcosi  —  gsini  }•  X 
-f  \  e  cos2  i  —  (/  -f-  h)  sin  i  cos  *  +  &  sin2 1  }•  F 
-j-  \  /cos2 1  -f  (e  —  &)  sin  *  cos  i  —  h  sin2  i\  Z 
-f-  §  cos  4  —  R  sin  t. 
Since  sin2  i  -f-  cos2  t  =  1,  and  by  substituting  1  —  sin2  i 

for  cos2  it  we  have 

P7  =  Y+  \  d  cos  i — g  sin  t  }>  JT-f «{  e — (/-f  A)  sin  t  cos  i  "I 

_  (g--^)sinH'  }•  Y+  \  f  +  (e  —  fysinicoai          f  (96) 

—  (/  +  A) sin**  }•  ^+  §cost—  jBsin*.  J 


HEELING  ERROR  169 

Equations  (92)  and  (96)  are  of  the  form 

X'  =  X  +  diX  +  biY  +  aZ  +  Pi ,  (97) 

Y'  =  Y  +  diX  +  e,Y  +  fiZ  +  Qt ,  (98) 

if 

di  =  0,. 

bi=  b  cos  i  —  c  sin  i. 
a  =  b  sin  i  -\-  c  cos  i. 
di  =  d  cos  i  —  a  sin  i. 

I       /QQ^ 

e»  =  e  —  (/  +  ft)  sin  i  cos  i  —  (e- —  Jc)  sin2  i.  '    ^     ' 


fi  =  f  -f-  (e  —  A;)  sin  t  cos  i  —  (f  +  ft)  sin2  i. 

Qt  =  Q  cos  i  —  E  sin  i. 

99.  The  coefficients  when  ship  is  heeled. — As  the  values  of 
the  parameters  and  magnets  have  changed  in  consequence  of 
the  ship's  heeling,  so  have  the  magnetic  coefficients  which  de- 
pend on  them. 

Therefore,  if  A* ,  2Ii ,  95i ,  (£* ,  ©< ,  (£* ,  respectively,  repre- 
sent the  altered1  values  of  the  coefficients  A,  2C,  SB,  (£,  2),  (£, 
due  to  the  heeling  of  the  ship  through  i°9  by  substitution  in 
the  equations  for  the  exact  coefficients,  we  have 

A<      =  A —  ^  Q      sin  i  cos  i „ — sin2  i.  (100) 

Ai5ti  =  A5T  cos  i  +  ~~^     s^n  *•  (101) 

=  ASB  +  -{  5  sin  i  —  c  versin  i  }•  tan  0.  (102) 

7?  1 

=  A(S  +  •{  (e  —  A;)  sin  t  cos  i  —  «-  sin  i 

ft)  sin2  O  tan  0      fi-wrint      l^^ 
=  A®  +  £i^  sin  t  cos  i  +  i^  sin2  i.  (104) 

=  A@  cos  i—   — t-^  sin  t.  (105) 

Approximate  values  of  the  coefficients  when  the  ship  is 
heeled. — If  the  soft  iron  is  symmetrically  arranged  on  each 


170  NAVIGATION — COMPASS  DEVIATION 

side  of  the  fore-and-aft  plane  through  the  compass,  1),  d,  f,  h, 
21  and  (£  will  be  zero;  and  as  a  steady  angle  of  heel  would  be 
small,  we  may  without  much  sacrifice  of  accuracy  replace  sin  i 
by  if  letting  cos  i  =  1,  versin  i  =  0,  and  sin2  i  =  0.  Equa- 
tions (100)  to  (105)  will  then  give 
\i  =  A. 


8«  =  9. 

R=<£+|(*-*-f.)tanrf.     :  K106) 

(£«  =  &  -f  Ji,  if  J  =  1. 1  c  — 
X 

Sh  =.  2). 


100.  Deduction  of  the  equation  expressing  heeling  devia- 
tion.— If  8  represents  the  deviation  for  a  given  compass  course 
z'  when  the  ship  is  on  an  even  keel,  Si  the  deviation  for  the 
same  compass  course  when  the  ship  heels  i°  to  starboard, 
then  equation  (34)  becomes  in  each  case,  the  approximate 
heeling  coefficients  being  substituted, 

8  (1  _  2)  Cos  2z')  =  £t  +  $  sin  2'  +  (£  cos  z'  -^  ©  sin  2z* 
+  (£  cos  2z' 

8,  (1  —  &  cos  2zr)  =  +^jri  +  %  sin  /+((£  +  «ft)  cos  s' 

+  S)  sin  2«'  —  cJr9  i  cos  2s'. 
2A 

Therefore,  since  by  the  hypothesis  21  and  (£  are  zero, 
(8,  _  8)   (i_S)  cos  2/)  =  ^=^  i  +  Jt  cos  it 


Substituting  cos2   2'  —  sin2   2'   for  cos   22',   multiplying 

XI  ^^__    >y 

A,-     i  by  (sin2  2'  +  cos2  z')?  rearranging  the  terms  in  the 


HEELING  ERROR  171 

right  member  of  the  equation,,  also  regarding  (8*  —  8)  2)  cos 
2zf  as  zero,  and  transposing  8  we  have 


Si  =  8  +  Ji  cos  /+  ~  i  sin2  z'  —  -|  t  cos2  z',      (107) 

which  will  give  the  heeling  deviation  on  the  compass  heading 
z'  when  8,  J ,  c,  and  g  are  known.  It  has  already  been  shown 
how  to  find  8  (Art.  55),  also  c  and  g  (Arts.  96  and  92).  The 
method  of  finding  J  will  be  explained  in  Art.  103. 

101.  General  effect  of  ship's  heeling  on  the  deviation  of 
the  compass. — Equation  (107)  shows  that  the  effect  of  heel- 
ing, besides  altering  the  term  (£  by  the  expression  Ji,  is  to  in- 
troduce a  constant  term  and  a  quadrantal  term  of  the  @?  type ; 
iron  which  is  symmetrical  with  the  ship  upright  becoming 
unsymmetrical  when  the  ship  heels.  It  is  readily  seen  that 
c  introduces  a  ( —  &)  =  ci,  and  gt  a  (+  d)  =  —  gi,  when  the 
ship  heels,  and  that  these  cause  the  2-U  and  @^.  c  represents 
vertical  soft  iron  in  the  midship  line,  as  the  smokestack;  the 
effect  of  c  depends  on  the  proximity  of  the  compass  to  the  pole 
of  c,  and  is  a  minimum  when  the  ship  heads  North  or  South, 

a  maximum  when  the  ship  heads  East  or  West,   -y  is  generally 

A 

-f-  and  seldom  exceeds  .100  for  the  usual  positions  of  compasses- 
The  parameter  g  represents  soft  iron  parallel  to  the  axis  of  X, 
as  the  keel  or  propeller  shaft;  the  effect  of  g  depends  on  the 
proximity  of  the  compass  to  the  pole  of  g,  being  greatest  when 
the  compass  is  well  forward  or  well  aft  and  the  ship's  head 
North  or  South.  The  effect  of  g  is  a  minimum  so  far  as 
location  of  the  compass  is  concerned  when  the  compass  is  at 
equal  distances  from  each  end  of  the  ship,  and  disappears  in 

all  cases  when  the  ship  heads  East  or  West,  -j  is  generally  + 
and  seldom  exceeds  .100. 

The  effects  of  both  c  and  g  may  be  neglected  in  the  ordi- 
nary cases  in  navigation. 


172  NAVIGATION  —  COMPASS  DEVIATION 

The  term  Ji  cos  z',  however,  cannot  be  neglected,  as  it  is 
often  large  in  amount. 

When  J  is  (  —  )  ,  as  it  usually  is  for  compasses  on  the  upper 
deck,  it  represents  a  deviation  of  the  North  point  of  the  needle 
to  windward;  when  +?  as  it  may  be  for  a  compass  on  the 
main  deck,  it  represents  a  deviation  to  leeward.  (  —  )  J  is 
called  the  heeling  coefficient  to  windward,  (  —  )  Ji  cos  z' 
being  the  heeling  error  to  windward,  a  maximum  on  North 
and  South  courses  and  a  minimum  on  East  and  West  courses. 
J  is  generally  a  fractional  number  and  indicates  the  heeling 
deviation,  for  each  degree  of  heel,  arising  from  a  change  in 
the  value  of  &  due  to  heeling,  on  N.  and  S.  courses  (p.  c.). 

102.  General  effect  of  the  ship's  heeling  on  the  coefficients 
determined  when  the  ship  is  upright.  —  An  inspection  of  equa- 
tions in  group   (106)   shows  that  the  coefficients  depending 
on  fore-and-aft  action,  93  and  2),  are  unaltered  ;  that  21  and  (£ 
undergo  a  slight  change;  and  that  &  is  considerably  altered. 
As  &  has  its  maximum  effect  when  the  ship  heads  North  or 
South  and  its  minimum  effect  when  the  ship  heads  East  or 
West,  it  is  apparent  that  the  heeling  error  is  a  maximum  or 
a  minimum  under  the  same  circumstances. 

103.  Different  ways  of  expressing  the  heeling  coefficient  and 

"1      /  7?  \ 

the  use  of  each.  —  The  expression  J  =  —  le  —  ~k  •  —  -~\    tan  B 

may  be  written  (  —  )  J  =  --  (  _  e  _|_  &  _|_  **  J  tan  6,  and 

then,  if  the  right  hand  member  of  the  equation  is  posi- 
tive, it  indicates  (as  is  usual  in  North  magnetic  latitude)  a 
deviation  of  the  North  end  of  the  needle  to  windward. 

y 

Since  tan  6  =  -T,  e  =  X  (1  —  ©)  —  1,  and  /x  —  1  =  ~k 


7?  1    / 

~2  ,  the  expression  —  J  =  j  f  — 


may 


HEELING  COEFFICIENT  173 

be  put  in  the  forms  below,  each  of  which  will  be  shown  to  serve 
a  special  purpose  : 


(a)—J=    x(— 

/M        J- 

~ 

(c]  —  J  =  (%)  +  1-  —  l)  tan  o  +  p~-^-  tan  0. 


Form  (a)  shows  the  changes  which  may  be  expected  in  (  —  )J 
on  change  of  magnetic  latitude  ;  \H  is  always  -f-  ;  tan  6  is  + 
in  the  northern,  (  —  )  in  the  southern  hemisphere  ;  at  the  usual 
position  of  a  standard  compass  on  a  ship  built  in  North  mag- 
netic latitude,  (  —  )  e,  +  Tct  and  +  R  are  positive.  (  —  )  e  and 
+  fc  will  change  sign  in  South  latitude,  +  R  will  not,  there- 
fore the  heeling  error  will  be  to  windward  unless  the  ship  is  so 
far  South  in  the  southern  hemisphere  that  (  —  e  ~\-  Jc)  tan  0 

r> 

is  greater  than  -^  . 

Form  (6)  shows  how  the  heeling  deviation  is  caused; 
(  —  )  £g  i  expresses  the  effect  due  to  vertical  induction  in 

horizontal  soft  iron  represented  by  the  rod  (  —  e)  ,  inclined  at 
an  angle  i  to  the  horizontal  plane,  and  acting  against  a  di- 
rective force  \H  ;  the  effect  being  a  heeling  error  to  wind- 
ward in  North  magnetic  latitudes,  to  leeward  in  South  mag- 

netic latitudes.     -  :  —  i  expresses  the  combined  effect  of 

vertical  induction  in  vertical  soft  iron  represented  by  the  rod 
-f-  &  and  the  component  to  keel  of  the  subpermanent  mag- 
netism represented  by  +  R,  both  acting  at  an  angle  i  from 
the  vertical  and  against  the  directive  force  XH.  The  effect  of 
this  part  is  a  heeling  error  to  windward  or  leeward  according 
as  the  resultant  force  of  JcZ  +  R  is  +  or  (  —  )  . 


174  NAVIGATION  —  COMPASS  DEVIATION 

Form  (c)  is  useful  for  computing  separately  the  heeling 
deviation  due  to  (1)  vertical  induction  in  horizontal  trans- 
verse soft  iron,  (2)  vertical  induction  in  vertical  soft  iron 

and  vertical  subpermanent  magnetism  ;  f  3)  +  -j  —  1  J  tan  0 

—  eZ  j  fj.—  l   .  JcZ+R 

=  -Try-  expressing  the  first  part,  and  {—  ^  —  tan  0  =         Jl 

expressing  the  second  part. 

Form  (d)  is  the  most  convenient  form  for  computing  the 
heeling  coefficient  ;  the  value  of  3)  having  been  determined  by 
analysis  of  a  table  of  deviations  or  from  observations  on  two 
headings,  p  and  A  by  vertical  and  horizontal  vibrations  on 
board  and  ashore,  and  6  taken  from  a  chart  of  magnetic  dip. 

In  order  that  there  may  be  no  semicircular  heeling  error 
(  —  )  J  must  be  zero  ;  therefore, 


[JL  -       =  6. 

That  is  to  say 

Mean  vertical  force  of  earth  and  ship 
/*=  —  i 

Vertical  force  of  earth 

and 

Mean  vertical  force  of  ship 
— 


Vertical  force  of  earth 


e. 


DETERMINATION  OF  HEELING  ERROR  175 

Ex.  18. — With  the  following  data  from  examples  11  and  12, 
viz.,  A  =  .8856,  3)  =  +  .05,  tan  0  —  2.86,  /x  =  1.0969,  g  — 
.1098,  c  being  neglected,  it  is  required  to  find  ( — )  J  and  the 
total  deviation  on  courses  South  and  NW.  (p.  c.)  when  the 
ship  is  heeled  1st  10°  to  starboard,  2d  10°  to  port.  Deviation, 
when  ship  is  upright,  on  South  +  4°  30',  on  NW.  — 16°. 

—  J=  f&  +  £  —  l\  tan  0  =  (.05  +  — ^-ggg 1J2.86  =  0.8254 

and  +  J=—  0.8254 

mi      ,.-,-,  j.       .       .  ,:\  +  if  heeled  to  starboard 

The  ship  heels  10°,  therefore  t  =  10  [•  '    .  , , 

J  —  if  heeled  to  port. 

At  South,  Ji  cos  z'  —  *L  i  cos2  z'  =  (— )  .8254  x  10  x  (—1)  —  .124  x  10  x  1 

A 

=  +  8°.  254  —  1°.24  =  +  7°. 014  =  +  7°  01'. 
At  NW.,  Ji  cosz'—  *L  i  cos2z'  =  (— )  .8254  x  10  x  .707— .124  x  10  x  .5 

A 

—  (_)  50.886  —  0°.620  =  (— )  6°.456  =  (— )  6°  27'. 

Ship  heeled  to  starboard,  on  South, 

8  =  +  4°  30'  +  (  +  )  7°  01'  =  +  11°  31'. 
Ship  heeled  to  port,  on  South, 

8  —  _p  4°  30'  —  (  +  )  7°  01'  =  (— )  2°  31'. 
Ship  heeled  to  starboard,  on  NW., 

8  =  (— )  16°  +  (— )  6°  27'  =  (— )  22°  27'. 
Ship  heeled  to  port,  on  NW., 

8  =  (— )  16°  +  (  +  )  6°  27'  =  (— )  9°  33'. 

104.  Determination  of  heeling  error  by  listing  and  then 
swinging  the  ship. — The  deviations  may  be  determined  by 
swinging  the  ship,  first,  upright,  then  heeled  i°.  The  dif- 
ference on  any  compass  azimuth  of  the  two  results  will  be  the 
heeling  error  for  that  angle  of  heel  on  that  course. 

The  results  of  this  practical  method  no  doubt  would  be 
more  satisfactory  after  the  work  was  done ;  it  is  a  tedious  pro- 
cess, however,  and  the  heeling  error  is  usually  determined 


176  NAVIGATION  —  COMPASS  DEVIATION 

theoretically  from   observations   already  shown  to  be   of  a 

simple  character,  when  not  corrected  by  the  tentative  method. 

If   swinging   takes   place   both   before   and   after   listing, 

c  may  be  found  from  the  observations  at  East  or  West,  as 

then  8i  =  S  -j  —  j-  i;  and  g  may  be  determined  from  the  obser- 

vations at  North  or  South,  as  then  8*  =  8  +  ^  —  y  *>  ^  and  J 
having  been  computed. 

105.  Correction  of  heeling  error  by  vibrations.  —  As  e  is 

minus  and  less  than  unity  when  the  quadrantal  deviation  has 
not  been  corrected,  it  is  thus  evident,  in  order  that  there  may 
be  no  heeling  deviation  under  such  circumstances,  that  the 
mean  vertical  force  at  the  position  of  the  compass  must  be 
less  than  that  on  shore,  and  that  the  time  of  n  vibrations  of  a 
vertical  needle  at  the  position  of  the  compass  represented  by 
T'  must  be  greater  than  the  time  of  the  same  number  of  vi- 
brations of  the  same  needle  ashore  represented  by  T. 
Eegarding  Ti  as  zero,  (63)  becomes 

jr-  =  fir,  =  **  +  g  cot  0  cos  * 

aQd  r  =  T  (108) 

COt  0COS2 


but  when  the  heeling  error  is  corrected  /u=  A.  (1  —  %))  =l-\-  e. 

T  T 

~~  */%  (!  —  $))  +  #  cot  6  cos  z  ~~  */l  +  e  +  gcotOcoaz' 

When  g  is  unknown,  the  ship's  head  may  be  placed  on  East 
or  West  magnetic, 


If  the  spheres  are  in  position  there  will  be  a  new  value  of  A 
and  perhaps  a  residual  value  of  S). 
Let  the  altered  values  be  A2  and  ®2  , 


HEELING  ADJUSTER  177 

Then  with  the  spheres  in  place  equations  (109)  and  (110) 
become 

T 

' 


It  is  thus  seen  that  the  heeling  error  may  be  corrected  by  so 
altering  the  vertical  force  that  the  vertical  vibrations  of  a  dip- 
ping needle  shall  take  place  in  the  proper  time  found,  accord- 
ing to  the  circumstances,  from  equations  (109),  (110),  (111), 
or  (112). 

The  vertical  force  is  altered  as  desired  by  the  vertical  move- 
ment of  a  vertical  magnet  in  the  binnacle  tube  below  the  cen- 
ter of  the  needle. 

It  is  customary  now  to  correct  the  heeling  error  by  what  is 
known  as  the  heeling  adjuster,  or  by  the  tentative  method 
(Art.  108  (5)). 

106.  Correction  of  heeling  error  by  using  the  heeling  ad- 
juster. —  The  heeling  adjuster  is  a  small  brass  box  provided 
with  levels  and  leveling  screws,  mounting  on  a  horizontal  axis 
a  needle  which  is  free  to  vibrate  in  the  vertical  plane,  its  ten- 
dency to  dip  being  counteracted  by  a  small  sliding  platinum 
weight  whose  distance  from  the  axis  of  suspension  may  be 
measured  by  a  scale  on  the  glass  cover.  There  is  a  small  glass 
window  in  each  end  provided  with  an  index  line  to  mark  the 
horizontal  plane.  Without  the  small  weight,  the  needle  before 
being  magnetized  was  exactly  balanced,  so  the  weight  is  in- 
tended to  balance  the  vertical  magnetic  force  ashore  or  on 
board.  Letting  b  and  a,  respectively,  denote  the  distance  be- 
tween the  weight  and  the  center  of  the  needle  when  the  needle 
is  exactly  balanced  on  board  and  ashore,  the  heeling  adjuster 

b      Z' 

being  properly  placed  in  the  magnetic  meridian,  then  —  =  ~^> 

and,  when  the  ship's  head  is  East  or  "West  magnetic,  —  =  p.. 


178  NAVIGATION — COMPASS  DEVIATION 

In  order  that  there  may  be  no  heeling  error  we  must  have 
p=l  +  e=-A(l  — ®). 

Therefore,  I  =  a\  (1  —  ©)  (113) 

before  the  quadrantal  spheres  are  placed. 

If  Ao ,  ®2  9  f*2  be  the  altered  values  of  A,  ®,  and  /*  after 
the  spheres  are  in  place, 

then  fji2  —  A2  (1  —  ©2)  or  &  =  aX2  (1  —  S>2).     (114) 

To  correct  the  error. — Place  the  weight  at  reading  &  from 
(113)  or  (114),  according  as  the  spheres  are  off  or  on  the 
brackets,  head  ship  East  or  West  magnetic,  put  the  adjuster 
on  the  brass  table  provided  for  this  purpose,  in  the  exact  posi- 
tion of  the  compass  needle,  the  adjuster  properly  placed  in 
the  meridian.  If  the  needle  remains  horizontal  there  is  no 
heeling  error.  If  one  end  dips,  place  the  heeling  corrector 
magnet  in  the  tube,  proper  pole  up;  raise  or  lower  it  till  the 
adjuster  needle  is  horizontal.  The  heeling  error  is  then  cor- 
rected. 

SECTION  VIII. 
COMPENSATION  OF  THE  COMPASS. 

107.  Principles  and  object  of  compensation. — It  has  been 
shown  that  each  kind  of  deviation  is  due  to  certain  forces, 
either  of  attraction  or  repulsion,  acting  on  the  North  point  of 
the  compass  needle,  and  it  is  evident  from  the  known  laws  of 
magnetic  action  that  these  forces  can  be  neutralized  and  hence 
deviation  reduced  to  zero,  by  introducing  other  forces  of  the 
same  magnitudes,  but  such  as  to  act  in  the  opposite  directions. 
By  compensation,  the  deviations  are  not  only  reduced  to  zero, 
or  to  convenient  amounts,  so  that  a  change  in  azimuth  of  the 
ship's  head  is  represented  by  a  similar  apparent  movement  of 
the  compass,  but  the  directive  force  of  the  needle  is  equalized 
on  the  different  headings.  After  compensation,  all  cor- 
rectors should  be  secured  in  place  and  their  positions  noted  in 
the  Compass  Journal. 


COMPENSATION  OF  THE  COMPASS  179 

108.  Order  of  compensation. — Since  the  correctors,  when 
in  place,  exert  a  mutual  action  on  each  other  and  thereby 
create  forces  additional  to  those  of  the  ship,  it  is  essential  that 
the  semicircular  correction,  which  is  the  largest  and  most  im- 
portant one,  should  be  made  when  the  magnetic  conditions 
approximate  as  nearly  as  possible  to  those  when  the  compen- 
sation is  complete;  therefore,  the  quadrantal  and  heeling 
correction  should  precede  the  semicircular  correction.  The 
quadrantal  spheres,  when  in  place,  correct  a  portion  of  the 
heeling  error  and  for  this  reason  it  is  desirable  that  the 
spheres  should  be  in  place  before  the  heeling  correction  is 
made.  However,  if  the  values  of  A.  and  ®,  before  the  spheres 
are  in  place,  are  known  by  computation,  the  heeling  correc- 
tion may  properly  be  made  by  the  method  of  Art.  106,  as 
the  first  correction,  the  distance  used  for  the  position  of  the 
weight  on  board  being  b  =  a\  (1  —  S)). 

If  the  correction  should  be  made  by  this  method  after  the 
spheres  have  been  placed,  we  must  find  the  distance  &  from 
the  equation  &  =  aX2  (1  —  S)2),  A2  and  ®2  being  altered 
values  of  A,  and  5D. 

The  heeling  error  may,  however,  be  corrected  by  the  tenta- 
tive method  and,  in  that  case,  the  following  will  be  the  order 
of  compensation : 

(1)  Correction  for  quadrantal  deviation  (approximate). 

(2)  Correction  for  heeling  error  (approximate). 

(3)  Correction  for  semicircular  deviation. 

(4)  Correction  for  quadrantal  deviation. 

(5)  Correction  for  heeling  error. 

(6)  Swing  for  residuals. 

It  being  assumed  that  the  ship  is  on  an  even  keel;  all 
movable  local  masses  of  iron  or  steel  in  the  vicinity  of  the 
compass  secured  in  their  normal  positions  for  sea;  and  the 
binnacle  of  Type  VI  stripped  of  all  correctors,  which  are  placed 
at  a  safe  distance ;  we  will  proceed  to  compensate  the  standard 
compass. 


180  NAVIGATION — COMPASS  DEVIATION 

From  data  by  computation. — Having  obtained  a  curve  of 
deviations  for  the  standard  compass  by  any  of  the  methods 
previously  referred  to,  take  from  the  Napier's  diagram  the 
compass  headings  corresponding  to  North,  NE.,  and  East 
magnetic,  then  head  the  ship  successively  on  those  rhumbs, 
steadying  on  each  at  least  four  minutes.  Note  carefully  the 
reading  of  the  steering  compass  when  the  ship  is  steadied  on 
those  rhumbs. 

Then  proceed  with 

(1)  The  approximate  correction  of  the  quadrantal  devia- 
tion.— If  the  value  of  S)  is  known  or  can  be  estimated,  place 
the  spheres  on  the  brackets  according  to  Table  III  of  "  Diehl's 
Compensation  of  the  Compass/'  or  Table  V  of  this  book. 

If  3)  is  unknown,  place  them  at  a  mean  position  of  13.5 
inches  for  the  7-inch  spheres.  If  spheres  of  this  size  overcor- 
rect  at  the  outer  limit,  use  smaller  ones,  remembering  that 
one  sphere  will  correct  half  as  much  as  two  of  the  same  size. 

(2)  The  approximate  correction  of  the  heeling  error. — 
Having  no  means  to  determine  A2  and  SD2 ,  place  the  ship's 
head  East  or  West  magnetic  by  means  of  the  steering  compass. 
The  needle  of  the  heeling  adjuster  having  been  made  hori- 
zontal on  shore,  with  the  weight  in  a  given  position,  must  be 
made   nearly  horizontal   on  board,   position   of  weight   un- 
changed, by  means  of  the  vertical  correcting  magnet  in  the 
central  tube,  the  non-weighted  end  inclined  perceptibly  up- 
wards. 

In  case  no  observations  were  made  ashore,  place  the  heel- 
ing magnet  in  its  tube,  North  end  up  in  North  magnetic 
latitude  unless  there  is  reason  to  know  that  the  ship's  verti- 
cal force  acts  upward,  and  lower  it  to  the  bottom. 

(3)  To  correct  the  semicircular  deviation. — Neglecting  the 
values  of  51  and  (£,  it  is  apparent  from  the  equation 

S3  sin  z  +  &  cos  z  +  ®2  sin  %z        •        i.  •  i.  **    • 

tan  8  =    ! ! • ,  m  which  ©2  is 

1  +  23  cos  z  —  (£  sin  z  +  ®2  cos  2z 


To  NEUTRALIZE  33  AND  (£  181 

the  coefficient  of  quadrantal  deviation  left  uncorrected, 
that  when  the  ship  heads  magnetic  North  or  South, 

tan  3  =  _-=^__-_-  •  that  the  forces   33  and  ®2  act  in  the 

meridian ;  that  the  transverse  force  (£  is  the  only  one  acting  to 
produce  deviation;  and  that  in  order  to  reduce  the  deviation 
on  those  headings  to  zero,  it  is  only  necessary  to  neutralize 
(£,  which  may  be  done  by  introducing  an  equal  but  opposing 
force  in  the  transverse  line. 

When  the  ship  heads  East  or  West  magnetic,  the  above 

[       rtl 

equation  becomes  tan  8  =  1  =P(£ SB  9  ^e  ^orces  ®  an^ 

SD2  act  in  the  meridian,  and  the  fore-and-aft  force  33  is  the 
only  one  acting  to  produce  deviation;  to  reduce  the  deviation 
on  those  headings  to  zero^  it  is  on^y  necessary  to  neutralize  33, 
which  may  be  done  by  introducing  an  equal  but  opposing  force 
in  the  fore-and-aft  line.  Therefore, 

To  neutralize  the  force  (£. — Head  the  vessel  North  magnetic 
and  keep  it  steady  by  the  steering  compass.  Eun  the  athwart- 
ship  carrier  down. 

If  the  compass  shows  easterly  deviation,  the  force  (£  draws 
the  North  point  of  the  needle  to  starboard ;  enter  one  or  more 
magnets  on  each  side  of  the  athwartship  carrier,  North  or 
red  ends  to  starboard;  move  the  carrier  up  or  down  until  the 
compass  points  North  magnetic. 

If  the  compass  shows  westerly  deviation,  the  force  (£  draws 
the  North  point  of  the  compass  needle  to  port;  enter  the 
athwartship  magnets  with  North  or  red  ends  to  port;  raise 
or  lower  the  carrier  till  the  compass  points  North  magnetic. 

Or  head  the  vessel  South  magnetic,  enter  the  athwartship 
magnets  with  North  or  red  ends  to  port  if  the  deviation  is 
easterly,  or  to  starboard  if  the  deviation  is  westerly;  raise  or 
lower  the  carrier  till  the  compass  points  South  magnetic. 

To  neutralize  the  force  33. — Head  the  vessel  East  magnetic 


182  NAVIGATION — COMPASS  DEVIATION 

and  keep  it  steady  by  the  steering  compass.  Kun  the  'fore- 
and-aft  carrier  down. 

If  the  compass  shows  easterly  deviation,  the  force  23  draws 
the  North  point  of  the  needle  to  head;  enter  one  or  more 
magnets  on  each  side  of  the  fore-and-aft  carrier,  North  or  red 
ends  forward;  move  the  carrier  up  or  down  till  the  compass 
heading  of  the  ship  is  East.  If  the  compass  shows  westerly 
deviation,  the  force  23  draws  the  North  point  of  the  needle  to 
stern ;  enter  the  fore-and-aft  magnets  with  North  or  red  ends 
aft  and  raise  or  lower  the  carrier  till  the  compass  heading  is 
East. 

Or  head  the  vessel  West  magnetic;  enter  the  fore-and-aft 
magnets  with  North  or  red  ends  aft,  if  the  deviation  is  east- 
erly, or  forward  if  the  deviation  is  westerly ;  raise  or  lower  the 
carrier  till  the  compass  heading  is  West. 

(4)  To  correct  the  quadrantal  deviation. — With  the  semi- 
circular forces  neutralized  there  remains  only  S)2  to  cause  de- 
viation, and  when  the  ship  heads  NE.,  SE.,  SW.,  or  NW.,  mag- 
netic, 2z  being  90°  or  270°,  the  equation  for  deviation  becomes 
tan  8  =  ±  5D2 ,  and  to  reduce  the  deviation  on  those  headings 
to  zero,  it  is  only  necessary  to  neutralize  the  force  $D2 ,  which 
may  be  done  by  introducing  an  equal  but  opposing  force. 

The  quadrantal  deviation  is  usually  positive,  and  hence  is 
corrected  by  placing  the  quadrantal  spheres  to  starboard  and 
port  of  the  compass  in  which  position  they  produce  a  negative 
quadrantal  deviation,  the  soft  iron  of  the  sphere  having  the 
effect  of  the  • —  a  and  +  e  rods ;  therefore, 

To  neutralize  the  remaining  quadrantal  force. — Having 
corrected  the  semicircular  deviation,  head  the  vessel  NE.  (or 
SE.,  SW.,  NW.)  magnetic  and  keep  it  steady  by  the  steering 
compass.  If  any  deviation  is  shown,  move  the  spheres  on  the 
side  brackets  in  or  out  until  the  compass  heading  is  NE.  (SE., 
SW.,  NW.). 

If  the   spheres   over-correct   at   the   outer   limits   of   the 


CORRECTION  OF  HEELING  ERROR  183 

brackets,  use  smaller  ones;  if  they  undercorrect  when  placed 
at  the  inner  limits,  use  larger  ones. 

(5)  To  correct  the  heeling  error. — In  case  the  heeling  cor- 
rector has  not  been  placed  by  shore  observations,  and  is  in  the- 
bottom  of  the  central  tube  of  the  binnacle,  if  there  is  sufficient 
sea  on  to  give  a  moderate  roll  on  a  North  or  South  course, 
steer  North  or  South  per  compass  and  observe  the  vibrations 
of  the  card  as  the  ship  rolls  from  side  to  side.     These  will  be 
greater  than  those  due  to  the  ship's  real  motion  in  azimuth 
when  the  heeling  error  is  material,  therefore  raise  the  heeling 
corrector  slowly  till  the  vibrations  almost  disappear,  leaving 
an  amplitude  of  1°  or  2°  to  avoid  over-correction.  It  must  not 
be  forgotten,  however,  that  the  correction  once  made  is  for 
that  particular  latitude  only  and  the  position  of  the  heeling 
corrector  must  be  changed  for  any  considerable  change  of 
magnetic  latitude. 

In  the  case  of  a  vessel  heeling  steadily  on  a  North  or  South 
course,  the  deviations  observed  when  heeled  may  be  compared 
with  those  when  the  ship  is  upright  on  the  same  course,  and 
the  difference  removed  by  raising  or  lowering  the  corrector. 

If  the  conditions  are  not  favorable  for  the  final  placing  of 
the  heeling  corrector,  reserve  it  for  a  future  time. 

(6)  Swing  ship  on  the  sixteen  principal  rhumbs  and  obtain 
a  table  of  residual  deviations;  either  readjust  the  correctors, 
proceeding  as  in  the  original  correction,  or  use  the  residual 
deviations  to  run  on.    In  case  of  re-compensation,  'the  vessel 
must  be  again  swung  for  a  final  table  of  deviations. 

The  -ship  may  now  be  placed  with  its  head  on  any  two  adja- 
cent cardinal  points  magnetic  by  the  standard  compass,  and 
the  other  compasses  corrected  for  semicircular  deviation ;  then 
on  the  intercardinal  point  magnetic  by  the  standard  compass, 
and  the  others  corrected  for  quadrantal  deviation. 


184  NAVIGATION — COMPASS  DEVIATION 

109.  Determination  of  the  magnetic  courses  when  compen- 
sating compasses  whose  deviations  are  unknown. — Select 
ahead  of  time  the  locality,  the  date,  and  the  limits  of  local 
apparent  time  between  which  the  observations  must  be  made. 

Take  from  the  Nautical  Almanac  the  sun's  declination  for 
the  instant  midway  between  the  time  limits,  and,  by  the  method 
of  Art.  58,  find  from  the  azimuth  tables  the  sun's  true  bearing 
at  intervals,  say,  of  ten  minutes  of  time,  for  the  known  latitude 
and  declination.  Apply  the  variation  for  the  locality  and 
obtain  the  magnetic  bearings.  Make  a  table  with  a  column  of 
magnetic  bearings  opposite  a  column  of  local  apparent  times, 
or  construct  a  curve ;  the  ordinates  representing  local  apparent 
times  at  intervals  of  ten  minutes ;  the  abscissae,  the  correspond- 
ing magnetic  azimuths  for  the  given  latitude  and  declination. 

On  the  date  selected  proceed  to  the  locality,  set  the  watch  to 
local  apparent  time,  and  shortly  before  the  first  selected  time, 
set  that  rhumb  of  the  pelorus  corresponding  with  the  desired 
magnetic  heading  to  the  ship's  head  and  clamp  the  plate.  Set 
the  sight  vanes  to  correspond  with  the  sun's  magnetic  bearing 
at  the  selected  time  and  clamp  the  vanes. 

Working  the  engines  slowly  and  using  the  helm,  bring  the 
sight  vanes  to  bear  on  the  sun  and  keep  them  there  till  the 
watch  shows  the  selected  local  apparent  time  when  the  ship 
heads  on  the  desired  magnetic  point,  and  the  ship's  head  per 
standard  should  be  noted,  also  the  ship's  head  per  steering 
compass.  Let  it  be  assumed  that  we  have  obtained  the  head- 
ings by  the  steering  compass  corresponding  to  any  two  adja- 
cent cardinal  points  and  the  intervening  quadrantal  point,  all 
magnetic,  and  that  a  careful  record  has  been  made  of  the 
same.  Then  to  compensate,  it  is  only  necessary  to  proceed 
as  explained  in  Art.  108. 


HEADING  MAGNETIC  COURSES  185 

Ex.  19. — Having  decided  to  compensate  the  compass  on 
April  18,  1905,  off  Cape  May,  in  latitude  39°  N".,  longitude 
74°  30'  W.,  between  the  hours  of  8  a.  m.  and  10  a.  m.,  local 
apparent  time,  it  is  required  to  make  a  table  of  magnetic 
bearings  of  the  sun  at  intervals  of  ten  minutes  between  the 
limits  named,  and  to  determine  the  compass  readings  corres- 
ponding to  magnetic  North,  NE.,  and  East.  Variation  —  8°. 

L.  A.  T.  of  middle  instant     9  00  00       O's  declination.  H.  D. 

Longitude  of  locality            4  68  OOW.  At  G.  A.  noon.      N.  10°  41'  12"  N.  52".46 

G.  A.  T.  of  middle  instant    1  58  00         Corr.                                      1  43  1.97 

or  April  18th.                      1.97                                  Dec.  =  N.  10°  42'  65"  103".36 

=  N.10°.7 

Lat.  39°  N.    I  On  page  90,  azimuth  tables. 
Dec.  10°  1ST.    f 

L.A.T.  9h     J  Z  =  N.I13°  32'  E. 
For  Dec.  11°  N.     Z  =  N.  112     34    E. 


Diff.  for  1°      of  Dec.     (— )  58' 
Diff.  for  0°.7  of  Dec.     (— )  41' 

Hence  we  have  for  Lat.  39°  K  and  Dec.  10°. 7  N".  as  follows : 

L.  A.  T.  Sun's  true  azimuth.     Sun's  magnetic  azimuth. 

8h  00m  a.  m.  100°  55'  108°  55' 

8    10    a.m.  102     44  110     44 

8    20    a.m.  104     36  112     36 

8    30    a.m.  106     33  114     33 

8    40    a.m.  108     34  116     34 

8    50    a.m.  110     40  118     40 


10    00    a.m.  128     32  136     32 

To  head  magnetic  North. — The  ship  being  on  the  station 
ahead  of  time — before  8  a.  m.,  local  apparent  time,  set  the 
North  point  of  the  pelorus  to  correspond  with  the  ship's  head, 


186  NAVIGATION — COMPASS  DEVIATION 

clamp  the  plate;  set  the  sight  vanes  to  the  magnetic  bearing 
of  the  sun  108°  55'  (by  table  or  curve)  and  clamp  the  vanes. 
So  manoeuvre  the  ship  by  the  engines  and  helm  that  the  sight 
vanes  will  point  directly  toward  the  sun.  By  helm  and  en- 
gines, keep  the  sight  vanes  on  the  sun  till  the  watch  set  to 
local  apparent  time  indicates  8  a.  m. 

At  that  instant  the  ship  heads  North  magnetic. 

The  standard  compass  reads  6°  (for  example). 

The  steering  compass  reads  8°  (for  example). 

Note  carefully  the  heading  by  the  steering  compass  at  this 
time. 

To  head  NE.  magnetic. — Say  it  is  desired  to  be  on  this 
heading  at  8h  20m  a.  m.  Set  the  NE.  point  of  the  pelorus  to 
the  ship's  head,  clamp  the  plate;  set  the  sight  vanes  to  the 
magnetic  bearing  of  the  sun  112°  36'  (by  table  or  curve) 
and  clamp  the  vanes.  Proceed  as  before,  keeping  the  vanes 
on  the  sun  till  8h  20m  a.  m.,  when  the  ship  heads  NE.  mag- 
netic and 

The  standard  compass  reads  38°  30'  (for  example). 

The  steering  compass  reads  22°  30'  (for  example). 

Note  carefully  the  heading  by  steering  compass. 

To  head  magnetic  East. — Let  8h  40m  a.  m.  be  the  selected 
time.  A  short  time  before  this  clamp  the  pelorus  plate  with 
the  East  point  on  the  forward  keel  line  or  indicator  and  clamp 
the  vanes  to  indicate  magnetic  bearing  of  the  sun  116°  34' 
(by  table  or  curve).  Proceed  as  before,  keeping  the  vanes  on 
the  sun  till  the  watch  set  to  local  apparent  time  shows  8h  40m 
a.  m.  At  that  instant  the  ship  will  be  heading  East  magnetic 
and 

The  standard  compass  reads  78°  00'  (for  example). 

The  steering  compass  reads  56°  00'  (for  example). 

Again  note  carefully  the  heading  by  steering  compass. 

To  head  a  magnetic  course  by  azimuth  circle. — Knowing 
the  magnetic  bearing  of  the  sun  for  a  given  instant  at  the 
place  of  observation,  or  of  a  distant  object,  set  the  direct  sight 


COMPENSATION  ON  ONE  HEADING  187 

vanes  of  the  azimuth  circle  to  the  right  or  left  of  the  ship's 
head  by  compass  by  an  angle  equal  to  that  which  the  sun  or 
object  is  to  the  right  or  left  of  the  magnetic  heading  desired 
at  the  selected  instant.  By  using  helm  and  engines  bring  the 
sight  vanes  on  the  sun,  keeping  them  on  it  till  the  watch  shows 
the  selected  time,  when  the  ship  will  be  on  the  desired  mag- 
netic heading.  In  the  case  of  a  distant  object  the  time  is  not 
considered. 

110.  To  compensate  on  one  heading,  as  when  riding  to  a 
tide,  in  a  dry  dock,  or  when  alongside  a  wharf,  etc.* — Having 
obtained  the  exact  coefficients  23,  (£,  and  S)  by  any  of  the 
methods  already  referred  to,  also  the  magnetic  heading,  and 
knowing  the  compass  heading  and  deviation,  compute  the 
deviation  due  to  each  coefficient  by  substituting  that  coefficient 
alone  in  the  equation, 

23  sin  z  +  &  cos  z  +  S)  sin  2z 
~  1  +  33  cos  z  —  (£  sin  z  -j-  ©  cos  2z 

then  find  what  should  be  the  compass  heading  as  each  amount 
is  successively  neutralized. 

If  the  value  of  A.  is  known,  make  observations  with  the  heel- 
ing adjuster  ashore  and  on  board,  finding  b  =.  a\  (1  —  ©) ; 
and,  neglecting  g,  place  the  heeling  corrector  magnet  in  place 
(Art.  106).  However,  if  the  values  of  A2  and  S)2  may  be 
determined  after  the  quadrantal  spheres  are  in  place,  first  put 
the  spheres  on  the  brackets  and  correct  the  quadrantal  devia- 
tion; then,  finding  &  =  a\2  (1  —  S)2),  place  the  heeling  cor- 
rector. 

Move  the  quadrantal  correctors  in  or  out,  keeping  them 
equally  distant  from  the  compass  needles,  till  the  amount  of 
deviation  due  to  2)  is  corrected. 

Then  correct  the  amount  of  deviation  due  to  that  force, 
$8  or  (£,  which,  for  the  ship's  heading,  is  more  nearly  at  right 
angles  to  the  direction  of  the  compass  needle.  Thus,  if  the 
ship's  head  is  more  nearly  North  or  South,  eliminate  the  de- 

*  For  procedure  in  special  cases,   as  when  heading  North  (E.,   S.   or  W.),   mag- 
netic, etc.,   see  Appendix  B. 


188  NAVIGATION  —  COMPASS  DEVIATION 

viation  due  to  (£  by  means  of  the  athwartship  magnets  first, 
and  then  eliminate  that  due  to  33.  If  the  ship's  heading  is 
more  nearly  East  or  West,  reverse  this  procedure,  eliminating 
first  the  deviation  due  to  33  and  then  that  due  to  G£j 

When  the  forces  33,  (£,  and  2)  have  been  neutralized,  com- 
pass and  magnetic  courses  should  be  the  same  (31,  (£,  zero). 

Ex.  20.'  —  In  example  14,  Art.  95,  the  following  coefficients 
were  found  for  a  standard  compass  by  observations  of  de- 
viation and  horizontal  force  on  two  opposite  headings,  viz.: 
33  =  (—  )  .0747,  (£  =  (—  )  .3142,  S>  =  +  .1211. 

It  is  required  to  find  the  deviation  due  to  each  force  when 
the  ship  heads  199°  30'  (p.  s.  c.),  dev.  +  25°  30',  and  the 
compass  heading  per  standard  as  each  is  successively  neutral- 
ized, the  ship  being  kept  steady  on  the  corresponding  magnetic 
heading  by  the  steering  compass. 

To  find  the  deviation  due  to  2), 

ta^=Tff^=±TTcr-1=+-1211and  {,=+6.64<16". 

To  find  deviation  due  to  33, 


To  find  deviation  due  to 


Therefore,  note  the  corresponding  heading  by  the  steering 
compass  and  keep  the  ship  steadied  on  that  heading,  in  case 
the  vessel  is  not  secured  in  dock  or  to  a  wharf.  Then, 

(1)  By  means  of  the  spheres  neutralize  the  force  +  SD> 
making  the  ship's  head  per  standard  compass  206°  24'  16". 

(2)  By  means  of  the  athwarthship  magnets,  North  or  red 
ends  to  port,  neutralize  the  force  (  —  )  (£,  making  the  compass 
headings  222°  20'  51". 

(3)  By  means  of  the  fore-and-aft  magnets,  North  or  red 
ends  aft,  neutralize  the  force  (  —  )   33,  making  the  compass 
heading  225°,  which  is  the  magnetic  heading. 

As  8£  —  +  2°  52'  28",  the  amount  of  error  is  only  about  13'. 

1It  found,  by  neutralizing  SB  and  ©  in  a  certain  order,  that  the  elimination  of 
one  force  leaves  the  other  at  a  small  angle  with  the  needle,  a  condition  unfavorable 
for  its  elimination,  consider  the  effect  of  a  reversal  of  that  order  with  a  view  to 
improving  conditions. 


THE  DYGOGRAM  189 

111.  Values  of  A  and  E  to  be  left  unconnected. — In  all 

cases  of  the  compensation  of  the  compass,  when  A  or  E  or  their 
algebraic  sum  is  as  much  as  1°,  the  amount  should  be  left 
uncorrected.  A  has  a  constant  value  and  sign  on  all  head- 
ings, the  quadrantal  deviation  represented  by  E  varies  as 
cos  2z'  and  changes  sign  at  East  and  West. 

Therefore,  if  compensating  semicircular  deviation  on  North 
or  South,  the  amount  to  be  left  uncorrected  for  A  and  E  would 
be  the  algebraic  sum  of  the  amounts  due  to  their  signs  by 
analysis ;  if  on  heading  East  or  West,  it  would  be  the  algebraic 
sum  of  the  amounts  due  to  A  with  sign  unchanged  and  to  E 
with  sign  changed  from  that  by  analysis. 

SECTION  IX. 

112.1  The  Dygogram;  Its  Construction,  Description,  and 
Use. — The  dygogram  is  one  of  the  graphic  methods  of  repre- 
senting the  deviations  of  the  compass  for  magnetic  headings; 
it  also  shows  the  horizontal  components  of  the  magnetic  force 
acting  on  the  compass  needle,  the  directions  in  which  they  act, 
and  the  deviations  produced  by  each  component  as  well  as 
the  total  deviation  for  any  magnetic  heading. 

The  word  "  dygogram  "  is  a  contraction  for  "  dynamo-gonio- 
gram,"  meaning  a  "  force  and  angle  diagram."  It  is  a  geomet- 
rical construction  fulfilling  the  conditions  of  the  general  ex- 
pression 

_  2t  +  33  sin  z  +  &  cos  z  +  ©  sin  2z  +  (£  cos  2z 
I  -f-  $8  cos  z  — :  (£  sin  z  +  S)  cos  2z  —  (5  sin  2z ' 
as  will  be  shown  further  on. 

To  construct  the  Dygogram  when  21,  23,  (£,  £>,  and  &  are 
known. — Navigators  of  the  U.  S.  naval  service  have  blank 

Art.  1121  taken  from  an  article  by  Comdr.  John  Gibson,  U.  S.  N., 
in  Proceedings  of  U.  S.  Navai  Institute,  Vol.  XX,  No.  3. 


190  NAVIGATION — COMPASS  DEVIATION 

forms  supplied  upon  which  there  is  a  vertical  scale,  OP,  rep- 
resenting unity,  which  is  divided  into  100  equal  parts,  and  by 
estimate  into  1000  parts;  and  an  arc,  with  0  as  a  center  and 
radius  equal  to  OP,  divided  into  degrees,  upon  which  devia- 
tions may  be  read  off.  When  no  blank  is  at  hand,  a  similar 
scale  may  readily  be  constructed.  In  all  cases  the  line  OP  is 
equal  to  unity  and  is  vertical,  and  at  P  there  is  a  horizontal 
line. 

Example.— Let  %=  +  .053,   23  =  +  .222,   (£  =  +  .220, 
S)  =  +  .226,  @  =  +  .063. 

For  reference,  see  Fig.  53. 

From  P  lay  off  PA  =  ^  to  the  right  if  ^  is  + ,  to  the  left  if  — . 
"      A       »      AE  =  $     "          "       "  (£  "  +,       "  »    — . 

»      ^       "     ED' =<£)  upwards       »  ©  "+,  (as  it  usually  is). 
»      7)'     "    IXJ?'  =  58         "  "  33  "  +  ,  downwards  if— . 

«      jj/    t<      j?/jyr  =  (£  to  the  right  «  (£  «  +,  to  the  left  if  -. 


With  A  as  a  center,  and  a  radius  equal  to  AD'  = 
describe  a  circle,  called  the  "  generating  circle."  From  2V 
draw  a  straight  line  through  D'  and  produce  it  until  it  inter- 
sects the  generating  circle  a  second  time,  which  point  mark  Q. 
The  point  Q  is  called  the  "  pole  "  of  the  dygogram  and  is  one 
of  the  necessary  points  to  have.  From  D'  produce  ND'  for 
the  distance  D'S  equal  to  D'N.  Take  a  straight-edge  of  paper 
of  sufficient  length  and  lay  it  down  on  .ZV$;  mark,  on  the 
edge  of  the  paper,  dots  opposite  the  points  N,  D',  and  8;  move 
the  paper  around  so  that  the  center  dot  moves  on  the  circum- 
ference of  the  generating  circle  and  with  the  edge  always 
passing  through  the  pole  Q;  by  means  of  pencil  dots  opposite 
the  end  marks  on  the  paper-edge,  a  sufficient  number  of  points 
may  be  obtained  for  drawing  in  free  hand  the  curve  of  the 
dygogram. 

To  mark  the  dygogram  for  magnetic  headings,  lay  a  pro- 
tractor on  the  line  NS,  its  center  at  Q,  and  dot  off  the  head- 
ings required  (usually  every  15°  rhumb)  ;  through  each  of 


ILLUSTRATING  CONSTRUCTION 


191 


30' 


FIG. 


192  NAVIGATION — COMPASS  DEVIATION 

these  points  and  through  Q  draw  a  line  and  extend  it  across 
the  dygogram.  Where  the  lines  cut  the  dygogram  are  the 
points  required;  the  first  cut  to  the  right  of  N.  (looking  from 
S.  to  N.)  is,  say,  15°,  the  2d,  30°,  the  3d,  45°,  the  4th,  60°, 
the  5th,  75°,  etc.  Draw  small  circles  around  the  points  and 
mark  each  one  correctly. 

To  construct  a  table  of  deviations  for  magnetic  headings 
when  only  the  exact  coefficients  are  known. — Having  pro- 
ceeded so  far  as  to  find  the  required  points  of  the  curve  and 
marked  each  correctly,  as  explained  above,  draw  a  line  from 
each  point  to  0  (or  until  it  intersects  the  graduated  arc)  ; 
the  deviation  then  for  each  magnetic  heading  is  shown  by  the 
angle  which  the  line  drawn  from  that  point  makes  at  0  with 
the  vertical  graduated  line,  and  is  read  off  from  the  graduated 
arc;  if  to  the  right  of  the  vertical  line  the  deviation  is  East 
or  -}-,  and  if  to  the  left,  West  or  — .  It  is  usual  to  record  the 
deviations  in  a  tabular  form  as  follows  (as  example,  case  of 
Fig.  53  is  taken)  : 

Magnetic  Heading.  Deviation.  Magnetic  Heading.  Deviation. 

0°  +13°  00'  180°  —  5°  30' 

15  +20  10  195  —  2  15 

30  +26  30  210  —  0  45 

45  +31  55  225  —  1  00 

60  +34  00  240  —  4  30 

75  +31  30  255  —  8  45 

90  +21  00  270  —12  00 

105  +  4  00  285  —13  15 

120  —  7  30  300  —11  15 

135  —11  50  315  —  7  00 

150  —11  30  330  —  1  00 

165  —  9  00  345  +5  50 

To  construct  a  table  of  deviations  for  compass  headings 
when  only  the  exact  coefficients  are  known. — In  practice  it 
is  necessary  to  have  a  table  of  deviations  for  "  compass  head- 
ings," and  to  get  it  when  only  the  exact  coefficients  are  known, 
proceed  as  explained  for  constructing  the  dygogram  and  the 
table  of  deviations  for  magnetic  headings.  Then  construct 
the  Napier's  curve  by  laying  off  on  the  Napier's  diagram  along 


THEORY  OF  THE  DYGOGEAM  193 

the  "full  lines"  the  deviations  for  the  magnetic  headings; 
draw  in  the  curve  and  then  take  off  the  deviations  along  the 
"  dotted  lines  "  for  the  "  compass  headings/'  and  record  them 
in  tabular  form,  one  column  being  "  Compass  Headings  "  and 
the  other  one  "  Deviations." 

To  show  that  the  dygogram  satisfies  the  conditions  of  the 
general  expression. — 

_  21  +  33  sin  z  +  (£  cos  z  +  ©  sin  2z  +  (£  cos  2z 

~  1  +  $  cos  z  —  (£  sin  z  +  &  cos  2z  • —  (£  sin  2z 

For  reference,  see  Fig.  54. 

Construct  the  dygogram,  as  previously  explained,  and  take 
any  point  R  of  the  dygogram  corresponding  to  the  magnetic 
heading  z.  The  position  of  the  different  coefficients,  or  the 
lines  representing  the  forces,  as  laid  down  in  constructing  the 
dygogram,  are  for  a  magnetic  heading  North;  for  any  other 
heading  z  the  lines  and  different  triangles  remain  of  the  orig- 
inal size,  but  assume  new  positions  in  regard  to  the  center  A. 
As  the  ship  swings  through  the  magnetic  azimuth  z,  the  keel 
line  DD'B'  swings  around  D  as  a  center,  through  the  angle  z, 
in  the  new  position  DD"K,  cutting  the  generating  circle  at 
the  point  D".  According  to  the  construction  of  the  dygogram 
a  line,  QD"R,  making  an  angle  z  at  Q  with  the  line  NS,  will 
cut  the  dygogram  at  the  point  for  the  azimuth  z;  this  line 
will  pass  through  D"  because  the  angles  D'QD"  and  D'DD" 
are  each  equal  to  z,  and,  as  both  Q  and  D  are  on  the  circum- 
ference of  the  circle,  the  angles  are  each  measured  by  half  the 
same  arc,  D'D".  According  to  the  construction,  D"R  =  D'N 
=  V93a+©a,and  by  geometry,  the  angle  B'D'N  =  KD"R  =  a; 
therefore,  a  perpendicular  let  fall  from  R  upon  DD"  produced 
will  cut  at  K  such  that  D"K  -=  »  and  KR  =  <£.  Thus  it  is 
seen  that,  in  swinging  through  an  azimuth  z,  the  triangle  of 
polar  forces,  D'B'N,  has  assumed  the  position  D"KR. 

The  triangle  A  ED'  will  revolve  around  A  as  a  center  in 


194  NAVIGATION — COMPASS  DEVIATIO' 

such  a  manner  that  while  the  ship  turns  through  an  angle  z, 
the  triangle  AED'  will  turn  through  an  angle  2z.  Above  it 
was  seen  that  half  the  arc  D'D"  measured  the  angle  D'QD"=z; 
therefore,  the  angle  at  the  center,  measured  by  the  same  arc, 
would  be  equal  to  2z;  that  is,  D'AD"  =  2z,  and  therefore  the 
other  sides  of  the  triangle,  AE  and  ED',  will  turn  through  the 
angle  2z;  or  EAEf  —  2z  and  E'D"D"f  —  2z  (Dm  being  verti- 
cally below  D"  on  the  line  PBC). 

The  forces,  as  represented  by  the  coefficients,  have  kept  their 
original  values  or  strength,  but  now  act  in  new  directions  to 
produce  deviation  and  to  affect  the  directive  force  of  the 
needle.  PA  =  $1  remains  constant;  AE'  =  (£,  but  acts  at 
the  angle  2z  with  its  former  position;  E'D"  =  ®  acts  at  an 
angle  2z;  D"K  =  33  and  KB  —  (£,  but  each  acts  at  the  angle 
z  with  its  former  position. 

From  each  of  the  points  E',  D",  K,  and  E  let  fall  perpen- 
diculars upon  the  two  axes  having  P  as  an  origin. 

As  R  is  the  point  of  the  dygogram  for  azimuth  z,  the  de- 

7"  /? 
viation  8  =  FOR;  then,  tan  8  =  Q-J*  in  which  LR  =  force 

of  earth  and  ship  to  magnetic  east  in  terms  of  mean  force  to 
"N.  as  unit ;  OL  =  force  of  earth  and  ship  to  magnetic  north  in 
terms  of  mean  force  to  1ST.  as  unit. 

By  referring  to  the  figure  it  is  seen  that  the  angles 
NQR,  D'DD",  D'QD",  DD"D"f,  D"KB,  KRL 
are  all  equal  to  each  other  and  to  the  azimuth  z.     It  may  also 
be  seen  that 

D'AD"  =  EAD'  —  EAD";  E'AS  =  E'AD"  —  EAD";  or 
EAD"  =  E'AD"  —  E'AS;  hence  D'AD"  =  EAD'  —  E'AD" 

+  E'AS. 
But 

EAD'=E'AD">;  .'.  D'AD"=E'AS  =  2z  =  MD"D'"  =  HE'S. 
Now,    LR  =  PC  =  PA  +  AS  +  SD'"  +  D'"B  +  EG; 
but  PA  =%;AS  =  ®cos2z. 


THEORY  OF  THE  DYGOGRAM 


«  =  4-  .040 

5b  =  •»•  .246 
S  =  +  .430 
J)  =  4-  .230 
«  -'  f  .114 


FIG.  54. 
GEOMETRICAL  DEMONSTRATION  OP  THE  DYGOGRAM. 


196  NAVIGATION  —  COMPASS  DEVIATION 

Let          E'M  =  r  and  MD"  —  s;  then  r  +  5  =  ®  ; 
SD'"  =  SM  +  MD'"  =  r  sin  2z  +  s  sin  2z; 
...  ##'"  =  (r  +  s)  sin  20  =  S)  sin  2s. 

ZTB  =  93  sin  2;  5(7  =  (£  cos  z; 

.-.  £#  =  51  +  93  sin  3  +  ©  cos  z  +.  ©  sin  2z  +  @  cos  22. 
Again,  OL  —  OP  +  (WG  —  FP)  +  (GJff  —  £#),  0 
being  the  point  where  the  horizontal  line  from  D"  cuts  the 
vertical  axis. 
But  OP  —  1  ;  WG  =  E'S  +  D'"D"  =r  cos  20  +  5  cos  2z 

=  (r  -\-  s)  cos  2z  =  ©  cos  2z. 
FP  =  @  sin  22;  OH  =  93  cos  0;  LH  =  &  sin  z; 
.-.  OL  =  1  +  93  cos  z  —  ©  sin  2  +  S)  cos  20  —  ©  sin  2z. 
51  +  93  sin  z  +  (£  cos  z  +  ®  sin  2z  +  ©  cos  2z 
" 


The  line  iV^  although  taken  as  the  zero  line  for  laying  off 
the  magnetic  headings,  does  not  represent  the  direction  of  the 
keel  line  of  the  ship  for  magnetic  North  or  South.  The  ver- 
tical line  represents  the  keel  line  for  magnetic  North  or  South 
(magnetic  meridian)  and  the  direction  for  any  other  heading  z 
is  represented  by  drawing  from  D  (vertically  below  E)9  a  line 
DD"K  making  an  angle  z  =  D'DD"  with  the  vertical.  As  93 
is  laid  off  to  head  and  &  to  starboard  (opposite,  if  negative), 
the  angle  B'D'N  =  KD"E  =  a  (the  starboard  angle). 

It  must  be  seen  that  the  points  marked  on  the  curve  of  the 
dygogram  are  not  really  directions  of  the  ship's  head,  but  are 
the  points  on  the  curve  which  show,  for  the  headings  desig- 
nated, the  deviations  of  the  compass  and  the  position  of  the 
forces  in  regard  to  the  meridian,  for  those  headings. 

To  represent  the  direction  of  the  ship's  head  on  the  dygo- 
gram for  any  designated  point  of  the  curve,  join  the  point  on 
the  curve  with  Q  and  note  the  intersection  of  this  line  with  the 
circumference  of  the  generating  circle;  a  line  drawn  from  D 
through  this  point  of  intersection  will  give  the  keel  line  of  the 


THEORY  OF  THE  DYGOGRAM  197 

ship.  Confusion  may  be  avoided  by  drawing  around  the  point 
of  intersection  with  the  generating  circle  the  outlines  of  a  ship 
with  its  head  in  the  proper  direction. 

OP  =  unity  =  mean  force  to  North  =  mean  directive  force 
in  the  compass  needle  =  \H.  Where  A  is  unity,  the  mean 
force  to  North  becomes  H,  the  horizontal  force  of  the  earth  at 
that  place. 

1  H' 

OL  =  -j-  -g~  cos  8  =  force  of  earth  and  ship  to  magnetic 

North,  in  terms  of  mean  force  to  North  as  unit  (for  any  par- 
ticular azimuth  z  of  ship's  head)  =  directive  force  of  needle. 

1    ft* 
LR  =-£-y  sin  S  =  force  of  earth  and  ship  to  magnetic 

East  (in  terms  of  XH,  the  mean  force  to  North  as  unit,  for 
any  particular  azimuth  z)  =  force  tending  to  draw  the  needle 
from  the  magnetic  meridian,  thus  causing  deviation. 

1  Hr 
OR  =-v-  TT-=  force  in  the  direction  of  the  disturbed  needle; 

the  needle  being  drawn  by  the  force  to  east  (LR),  out  of  the 
meridian,  through  the  angle  FOR  —8  for  that  particular  azi- 
muth z. 

The  angle  POA  =  deviation  due  to  constant  force  5t  (same 
for  all  headings). 

AOE'  =  deviation  due  to  induced  force  in  unsymmetrical 
soft  iron,  represented  by  coefficient  (£. 

E'OD"  =  deviation  due  to  induced  force  in  symmetrical 
soft  iron,  represented  by  coefficient  ©. 

D"OK  =  deviation  due  to  polar  force  to  head,  represented 
by  coefficient  23. 

KOR  =  deviation  due  to  polar  force  to  starboard,  repre- 
sented by  coefficient  (£. 

Of  course,  the  sum  of  any  two  or  more  of  these  angles  is 
equal  to  combined  deviation  caused  by  the  combined  forces 
designated.     Thus   the   deviation    for   magnetic    azimuth   z 
caused  by  the  forces  represented  by  5T,  (£,  and  S)  is 
POD"  =  POA  +AOE'  +  E'OD". 


198  -NAVIGATION — COMPASS  DEVIATION 

A  correct  idea  of  what  the  dygogram  is  may  be  obtained 
from  the  following,  viz. :  Suppose  a  compass  needle  pivoted 
at  0  (see  Fig.  54),  its  half  length  when  equal  to  OP  being 
considered  as  unity,  that  is,  equal  to  the  mean  force  to  North, 
XH.  Suppose  the  needle  capable  of  assuming  a  length  pro- 
portional to  the  force  in  the  direction  of  its  length,  for  each 
heading.  From  an  inspection  of  the  dygogram,  it  is  seen  that 

(-I         TTf  \ 
OR  =  ~j-  ~  j 

varies  in  amount  or  length  as  the  ship  swings  in  azimuth. 
Now,  as  the  ship  swings  in  azimuth,  through  a  complete  circle, 
the  end  of  the  needle  will  trace  out  the  curve  of  the  dygogram, 
its  end,  at  any  azimuth  z  being  at  the  point  R  of  the  dygogram, 
showing  a  deviation  8  =  FOR. 

Various  cases  might  be  given  where  a  knowledge  of  the 
dygogram  would  be  of  great  assistance;  such  as  where  the 
values  of  2t,  23,  (£,  £>,  and  (£  are  all  known,  and  it  is  desired 
to  compensate  on  any  heading  while  at  the  dock ;  in  which  case 
the  deviation  due  to  ST,  (£,  23,  and  (£  has  to  be  left  uncorrected 
in  compensating  the  quadrantal  deviation,  and  that  due  to 
51  and  @:  left  uncorrected  in  compensating  that  due  to  23  and  (£. 

Fig.  55  shows  the  manner  in  which  the  various  forces  re- 
volve, by  which  the  final  curve  of  the  dygogram  is  traced  out, 
when  all  the  forces  have  appreciable  values  as  represented  by 
the  exact  coefficients. 

OP  is  equal  to  unity  =  \H. 

PA  is  the  dygogram  due  to  the  constant  force  represented 
by  .ST. 

The  inner  circle  is  the  dygogram  due  to  the  induced  force 
represented  by  @?,  standing  to  one  side  of  the  meridian  line  on 
account  of  the  constant  force  21 ;  the  circle  is  properly  marked. 

The  next  circle,  having  a  radius  equal  to  V  ®2  +  ®2>  ig  ^ne 
dygogram  due  to  both  induced  forces,  represented  by  (£  and  ©, 
standing  to  one  side  of  the  meridian  line  on  account  of  the 
constant  force  21,  and  if  21  is  zero  its  center  is  at  P. 


THEORY  OF  THE  DYGOGEAM 


199 


200  NAVIGATION — COMPASS  DEVIATION 

The  small  shaded  triangle  is  the  triangle  of  induced  forces 
producing  the  quadrantal  deviation,  and  revolves  around  A  as 
center,  the  rate  of  revolution  being  double  that  of  the  ship  in 
swinging. 

The  large  shaded  triangle  is  the  triangle  of  polar  forces  pro- 
ducing the  semicircular  deviation;  it  revolves  on  the  circum- 
ference of  the  quadrantal  circle,  its  apex  continually  touching 
that  of  the  inner  triangle,  the  center  of  revolution  being  at  the 
point  D  (see  Figs.  53  and  54)  and  its  rate  of  revolution  being 
the  same  as  that  of  the  ship  in  swinging. 

The  final  curve,  that  traced  out  by  the  outer  corner  of  the 
triangle  of  polar  forces,  is  the  curve  of  the  dygogram. 

If  both  21  and  @:  are  zero,  the  center  of  the  second  circle 
becomes  P  and  its  radius  3). 

If  51,  @?,  and  2)  are  all  zero,  the  dygogram  for  the  semi- 
circular forces  is  a  circle  whose  center  is  P  and  radius 
V  332  +  ®2 ;  if  then  either  S3  or  (£  becomes  zero,  the  dygogram 
of  the  remaining  force  will  be  a  circle  whose  center  will  be  P 
and  whose  radius  will  be  the  remaining  force. 

113.  Given  the  deviations  and  the  horizontal  force  on  two 
courses,  regarding  51  and  @  as  zero,  to  find  A,  $,  (£,  and  $  by 
construction. 

Let  Z-L  and  Z2  be  the  two  magnetic  courses,  8^  and  82  the 

TTt  TTt 

corresponding  deviations,  -rr  and  -~^  be  the  horizontal  forces 

in  terms  of  H. 

1   H' 

Eeferring  to  Fig.  54,  it  is  seen  that  if  OR  =  y  -go  then 

OP  =  1,  OL  =  1  +  $  cos  z  —  &  sin  z  '+  £>  cos  2z  —  @  sin  2zf 

and  LR  =  51  +  SB  sin  z  +  (£  cos  z  +  %)  sin  2z  +  ©  cos  2z. 

iff 
Now,   if   OR  is   taken   as  ^- ,  we   shall   have    OP   =   \, 

OL  —  A  +  A$  cos  z  —  A(£  sin  z  +  A®  cos  2z  —  A©  sin  2z,  and 
LR  =  A5C  +  A33  sin  3  +  AS  cos  a  +  A®  sin  2z  +  A(£  cos  2z. 


OBSERVATIONS  ON  Two  COURSES 


201 


Points  on  a  dygogram  corresponding  to  these  last  values  of 
OL  and  LR  as  coordinates  may  be  found  thus :  For  R^  lay  off 

the  angle  POR±  =  8±  and  take  OR,=^  for  R2  lay  off  the 

angle  POR2  =  32  and  take  OR2  =  —?.     Call  R,  and  R2  datum 
points  (Figs.  56  and  57). 


FIG.  57. 

There  are  two  different  constructions  under  the  above  gen- 
eral heading. 

(1)  When  the  two  magnetic  courses  are  diametrically  op- 
posite (Fig.  56). 

Find  the  datum  points  R.L  and  R2  and  draw  RJ&2 ,  bisecting 
it  in  Q-,  a  point  of  the  generating  circle.  Through  G  draw 
AD,  parallel  to  the  magnetic  direction  of  the  keel  line,  and 
intersecting  OP  in  D.  Draw  OD'  perpendicular  to  AD  inter- 
secting OP  in  D'.  Bisect  DD'  at  P  and  drop  a  perpendicular 


202  NAVIGATION — COMPASS  DEVIATION 

from  Ri  and  Rzon  AD.  Then  OP  —  A,  GA  ==  GA'  —  A93, 
AS,.  =  A'R2  =  A®,  and  PD'  =  AS).  By  drawing  the  outline 
of  a  ship  about  G,  the  signs  of  A93  and  AS  become  apparent. 
Having  these  quantities,  find  93,  (£,  and  ®  and  construct  the 
dygogram. 

If  the  two  magnetic  courses  are  !NT.  and  S.,  G  will  be  at  D'. 
and  we  can  not  determine  D,  and  hence  neither  A  nor  S).  If 
they  are  E.  and  W.,  G  will  be  at  D,  and  we  can  not  determine 
Dr,  and  hence  neither  A  nor  3). 

(2)  When  the  two  magnetic  courses  are  not  diametrically 
opposite. — It  has  been  shown  in  Art.  112  that  a  point  of  the 
dygogram,  for  example  R,  Fig.  54,  may  be  found  by  laying  off 
the  angle  z  ==•  D'DD"9  then  measuring  off  D"K  =  93  and 
KR  =  (£.  From  Fig.  57  it  is  seen  that  R±  may  be  found  by 
laying  off  93  —  D'B  (=.  D"K),  making  the  angle  MD'B  =  z, 
and  DC  =  &  at  right  angles  to  the  course  line  D'B,  and  then 
completing  the  parallelogram. 

Now  for  another  course  MD'B'  lay  off  D'Br  —  93  in  the 
direction  of  the  course  line  and  DC'  =  &  at  right  angles  to  it, 
completing  the  parallelogram  and  finding  the  datum  point  R2. 

Hence  if  R±  is  a  datum  point  corresponding  to  one  course 
MD'B,  CR±  is  a  line  parallel  to  and  R^B  a  line  perpendicular 
to  the  course  line  through  that  datum  point. 

In  the  same  way  C'R2  and  R2B'  are  lines  through  R2  respec- 
tively parallel  to  and  perpendicular  to  the  second  course  line. 

Let  u  be  the  intersection  of  the  two  lines  through  R±  and  R2 
parallel  to  the  course  lines,  that  is  of  CR±  and  C'R2 ;  and  let 
v  be  the  intersection  of  lines  perpendicular  to  them,  that  is  of 
R±B  and  R2B' ;  then  from  geometry  it  is  plain  that  if  we 
draw  vD'  and  uDJ  they  will  intersect  at  w,  a  point' of  the  gen- 
erating circle  DD',  that  vD'  bisects  BvB'  and  uD  bisects  CuC', 
and  further  that  w  is  on  the  circle  passing  through  u,  v,  and 
the  datum  points  R^  and  R2. 

Therefore,  to  construct  a  dygogram  when  the  two  courses  are 


OBSERVATIONS  ON  Two  COURSES  203 

not  diametrically  opposite,  find  the  datum  points  R±  and  R2 
as  in  case  (1).  Through  R^  and  R2  draw  lines  parallel  to  the 
keel  lines  meeting  in  u  and  lines  perpendicular  thereto  meet- 
ing in  Vj,  the  keel  line  through  R±  corresponding  to  the  course 
Z-L  and  that  through  R2  to  the  course  z2. 

Through  v  draw  a  line  parallel  to  one  bisecting  the  angle 
between  course  lines,  in  other  words  bisecting  the  angle  R±uR2. 
This  line  cuts  the  vertical  line  in  D'.  From  u  drop  a  perpen- 
dicular on  vD',  intersecting  it  at  w  and  the  vertical  line  at  D. 
Bisect  DD'  at  P.  From  D  drop  perpendiculars  DO  and  DC', 
respectively,  on  the  1st  and  2d  course  lines ;  from  If,  perpen- 
diculars D'B  and  D'B',  respectively,  on  lines  at  right  angles  to 
said  course  lines. 

Then  OP  =  A,  and  DD'  is  the  diameter  of  the  generating 
circle. 

PD  =  AS),  D'B  =  D'B'  =  A33,  DC  ==  DC'  =±  Ad. 

Having  found  these  values,  find  the  coefficients,  and  con- 
struct the  dygogram. 

The  construction  fails  when  the  magnetic  courses  are 
equally  distant  from  any  cardinal  point ;  for,  if  equally  distant 
from  1ST.  or  S.,  RJ)  —  R2D  and  RJ)R2  —  the  difference  of 
magnetic  azimuths  =  R^wR2 ,  the  point  v  will  be  in  the  vertical 
line,  w  will  be  at  D,  and  it  will  be  impossible  to  determine  D' ; 
if  equally  distant  from  East  or  West,  w  will  coincide  with  D't 
and  it  will  be  impossible  to  determine  D* 

113a.  To  find  23  and  (£  from  observations  on  one  heading 

TJf 

(see  Art.  94).    Having  determined  8,  z,  and  ^|L  and  assuming 

A  and  ®,  let  OP  =  unity  represent  the  direction  of  magnetic 
North  (Fig.  56).  Describe  a  circle,  center  P,  radius  3),  cut- 
ting OP  in  D.  Draw  DA,  making  PDA  ==  z  and  cutting  the 
circle  in  G.  Lay  off  POR±  —  8±  (the  given  deviation  8),  to 

right  if  +,  to  left  if  (— )  ;  take  OR^  --\-^'>  dr°P  RiA 
perpendicular  to  DA;  then  GA  =  33  and  AR±  =  S.  An 
outline  of  a  ship  around  G,  heading  properly,  will  make  the 


CHAPTEE  V. 

PILOTING.— FIXING  SHIP'S  POSITION  NEAR  LAND.— 

DANGER  ANGLE.— DANGER  BEARINGS.— 

FOG   SIGNALS. 

114.  Piloting  in  its  broad  sense  is  the  act  of  conducting  a 
ship  where  navigation  is  dangerous,  as,  when  coasting,  passing 
through  channels,  and  into  harbors.  Before  reaching  pilot 
waters,  a  navigator  should  study  the  charts  and  sailing  direc- 
tions of  the  region,  know  that  they  are  up-to-date,  be  con- 
versant with  landmarks  and  aids  to  navigation,  and  the  state 
of  tides  and  currents  of  the  locality  at  the  time  when  he  may 
navigate  the  waters.  After  reaching  pilot  waters,  a  keen  look- 
out must  be  kept  for  dangers  as  well  as  aids  to  navigation, 
soundings  should  be  taken  and  depths  and  character  of  bot- 
tom obtained  compared  with  indications  of  the  chart ;  in  shoal 
water  the  hand  lead  should  be  kept  going.  Advantage  should 
be  taken  of  the  first  opportunity  to  locate  the  ship's  position 
by  bearings  of  known  objects,  and  having  laid  a  course  clear  of 
all  dangers,  the  ship's  position  must  be  frequently  plotted  by 
the  most  convenient  of  the  methods  herein  explained. 

Before  proceeding  to  explain  them  it  will  be  well  to  define 
general  terms  of  frequent  use  in  navigation. 

The  bearing  of  an  object  from  a  ship  is  the  angle  between 
the  meridian  and  the  great  circle  which  passes  through  the 
object  and  the  observer  on  board,  and  it  indicates  the  direc- 
tion in  which  the  object  is  seen  from  the  ship. 

It  is  called  true,  magnetic,  or  compass  according  as  the 
meridian  considered  is  that  passing  through  the  geographical 


FIXING  POSITION  NEAR  LAND  205 

poles,  the  magnetic  meridian,  or  the  direction  of  the  compass 
needle. 

On  board  ship  bearings,  by  compass  or  peloms,  are  measured 
from  North,  to  the  right,  from  0°  to  360°. 

A  line  of  position  is  any  line,  straight  or  curved,  on  which 
the  ship's  position  is  known  to  be.     It  is  obtained  from  ob- 
servations of  either  celestial  bodies,  or  terrestrial  objects. 
'  A  line  of  bearing. — When  a  line  of  position  is  obtained 
from  a  bearing,  to  make  it  more  distinctive  and  to  indicate ' 
its  origin,  it  is  called  a  line  of  bearing. 

Position  point. — Any  point  on  either  a  line  of  position  or 
a  line  of  bearing  at  which  the  ship's  position  may  be  assumed 
is  a  position  point;  the  actual  position  of  the  ship,  or  a  fix, 
is  determined  by  the  intersection  of  two  lines  of  bearing,  two 
lines  of  position,  or  one  of  each. 

115.  To  fix  the  position  of  the  ship  near  land  when  two 
or  more  landmarks  of  known  position  are  in  sight. 

(1)  By  sextant  angles. — Select  three  objects  so  as  to  give 
well-conditioned  circles  (see  Art.  34).  Generally  speaking, 
the  angles  should,  if  possible,  be  over  30°  and  the  objects 
in  line,  or  the  middle  one  nearest  the  observer.  Observe  by  a 
sextant,  whose  I.  C.  is  known,  the  angle  between  the  middle 
and  right  objects,  and  at  the  same  time  the  angle  between  the 
middle  and  left  objects,  which  are  known  respectively  as  the 
right  and  left  angles ;  set  the  right  and  left  arms  of  a  station 
pointer,  or  3-arm  protractor,  for  their  respective  angles,  place 
protractor  on  the  chart,  move  it  over  chart  till  the  beveled 
edge  of  each  arm  passes  simultaneously  each  through  its  own 
object.  The  center  of  the  instrument  locates  the  ship's  posi- 
tion on  the  chart. 

A  special  application  of  this  method  is  when  two  of  the 
objects  are  in  range  and  but  one  angle  is  taken. 

This  is  by  far  the  preferable  method  from  the  standpoint  of 
accuracy,  as  the  position  is  independent  of  compass  errors, 


206 


NAVIGATION 


speed  errors,  or  current  effect.  Angles  can  be  taken  from 
any  part  of  the  ship  whilst  bearings  must  be  taken  from  the 
standard  compass  or  pelorus. 

The  geometrical  theory  of  the  method  will  be  understood 
from  Fig.  58  which,  by  construction,  embodies  a  method  that 
can  be  used  in  the  absence  of  a  3-arm  protractor  or  tracing 
paper  on  which  the  angles  could  be  ruled.  Let  A.,  B,  and  C, 
Fig.  58,  be  the  three  known  positions.  The  observer  at  P 
measures  angle  x  between  A  and  B,  and  angle  y  between  B 


FIG.  58. 


and  C.  Using  the  two  objects  A  and  B  and  angle  x  alone,  the 
observer  may  be  anywhere  on  the  segment  APB  which  be^ 
comes  a  line  of  position.  If  x  is  <  90°,  the  line  of  position 
is  greater  than  a  semicircumference,  if  x  =  90°,  the  line  of 
position  is  180°,  if  x  is  >  90°,  the  line  of  position  is  <  180°, 
and  the  position  point  may  be  anywhere  on  the  arc,  since  all 
angles  on  the  same  segment  of  a  circle  equal  each  other. 

Using  the  two  objects  B  and  C  and  the  angle  y  alone,  the 
observer  may  be  anywhere  on  the  segment  BPC  which  becomes 
a  second  line  of  position,  the  length  of  which  is  governed  by 
rules  as  already  explained  for  the  first  line  of  position  ABPt 


BY  CROSS  BEARINGS  207 

and  the  position  point  is  somewhere  on  the  arc  BPC,  and. 
being  also  on  ABP,  is  at  their  intersection  P;  in  other  words, 
P  is  the  fix. 

There  are  three  cases  in  this  method:  (1)  Both  angles 
<  90°,  (2)  both  >  90°,  (3)  one  <  90°  and  one  >  90°. 
When  an  angle  is  <  90°,  the  center  of  the  circle  passing 
through  the  observer's  position  and  the  two  objects  is  on  the 
same  side  of  the  line  joining  the  two  points  as  the  observer ;  in 
this  case  lay  off  AD  and  BD  making  angles  90°  — x  with  AB, 
D  will  be  the  center  of  the  first  circle.  When  an  angle  is 
>  90°,  the  center  of  the  circle  passing  through  the  observer's 
position  and  the  two  objects  is  on  that  side  of  the  line  join- 
ing the  two  objects  remote  from  the  observer ;  in  this  case  lay 
off  BE  and  CE  making  angles  y  —  90°  with  BC,  E  will  be  the 
center  of  the  second  circle.  Hence,  we  have  the  general  rule : 
take  the  complement  of  the  observed  angle;  if  -{-,  the  center 
of  circle  will  lie  on  the  same  side  of  line  joining  observed  ob- 
jects as  the  observer;  if  ( — ),  the  center  of  circle  will  be  on 
the  opposite  side. 

The  indeterminate  case  is  when  the  observer  and  the  three 
objects  are  on  the  same  circle,  or  nearly  so,  or  when  the  two 
centers  nearly  coincide.  To  avoid  such  a  condition  see  Art.  34. 

In  case  no  protractor  is  at  hand  it  is  only  necessary  to 
measure  the  distance  AB,  erect  a  perpendicular  at  its  middle 
point  M,  and  lay  off  MD  =  MB  tan  (90°  —  x)  to  find  the 
center  D,  then  describe  the  circle  with  radius  DB.  In  a 
similar  way  find  E  and  with  radius  EB  describe  the  second 
circle,  the  intersection  giving  the  fix  P. 

(2)  By  cross  bearings. — This  consists  in  finding  the  fix 
by  two  or  more  lines  of  bearing.  The  bearings  per  standard 
compass  of  two  points  of  land,  or  objects,  whose  positions  are 
projected  on  the  chart,  having  been  obtained,  are  corrected  for 
the  deviation  due  to  the  ship's  head  at  the  instant  the  bear- 
ings were  taken.  Lay  the  parallel  rulers  on  the  nearest  com- 


208  NAVIGATION 

pass  rose,  the  edge  passing  through  the  center  and  the  degree 
on  the  circumference  representing  the  magnetic  bearing  of  a 
given  object,  transfer  the  edge  of  rulers  parallel  to  itself  till 
it  passes  through  the  given  object.  Draw  a  light  line  along 
the  edge.  This  is  a  line  of  bearing  and  the  position  of  the  ship 
is  somewhere  on  it.  In  a  similar  way  draw  a  second  line  of 
bearing  through  the  second  object,  and  the  ship  being  also 
somewhere  on  this  line,  the  fix  is  at  the  intersection  of  the 
two.  If  the  compass  rose  had  been  true  instead  of  magnetic, 
the  compass  bearing  would  have  been  corrected  for  variation 
as  well  as  for  the  deviation  of  the  compass.  The  difference 
of  bearings  should  be  as  near  90°  as  possible  for  best  results; 
if  the  difference  is  small,  15°  to  20°,  a  small  error  in  the' 
bearing  will  make  the  fix  uncertain.  The  position  determined 
from  the  bearings  of  only  two  objects  may  be  in  error,  even 
when  the  angles  of  intersection  are  good,  due  to  an  error  in 
the  assumed  deviation  or  even  an  error  of  the  chart ;  for  these 
reasons  a  third  line  of  bearing  should  be  obtained,  if  a  third 
object  is  available.  Should  these  three  lines  of  bearing  form 
at  their  intersection  only  a  small  equilateral  triangle,  its 
center  may  be  regarded  as  the  fix. 

In  finding  a  fix  by  cross  bearings,  take  first  the  bearing  of 
that  object  nearest  the  fore-and-aft  line,  ahead  or  astern,  since 
such  an  object  will  change  its  bearing  less  than  one  more 
nearly  abeam,  in  the  interval  between  bearings.  If  one  ob- 
ject is  ahead  or  abeam,  knowing  the  course,  its  bearing  is 
taken  mentally,  and  it  is  only  necessary  to  get  a  bearing  of 
the  second  object.  A  bearing  of  one  object  and  a  bearing 
of  a  range,  or  a  bearing  of  one  object  in  connection  with  a 
sextant  angle  to  another  object,  provided  the  angle  to  the 
second  object  is  sufficiently  large,  will  give  a  good  fix. 

116.  (B)  When  one  object  only  is  available. — Having 
taken  a  compass  bearing  of  the  single  object,  we  have  a  line 
of  bearing  which  is  true  or  magnetic  according  as  the  com- 


To  FIND  DISTANCES  .  209 

pass  bearing  is  corrected  for  compass  error  or  for  deviation 
alone.  If  true,  it  is  laid  down  from  a  true  rose ;  if  magnetic, 
from  a  magnetic  rose.  The  ship  is  somewhere  on  this  line. 
A  fix  on  this  line  may  be  gotten  from  its  intersection  with  a 
line  of  position  found  at  the  instant,  or  from  one  brought  up 
to  the  instant  of  getting  a  bearing  by  the  run,  or  by  knowing 
latitude  or  longitude,  or  by  knowing  the  distance  of  the  object. 

The  distance  of  the  object  may  be  estimated;  or  gotten 
from  its  angular  altitude,  if  its  height  is  known;  or  by  Buck- 
ner's  method  when  available;  or  by  an  accurate  range-finder.* 
Knowing  the  distance,  the 
ship  is  somewhere  on  a  line 
of  position  whose  center  is 
the  object  and  whose  radius 
is  the  distance,  and,  being 
also  on  the  line  of  bearing, 
the  ship  is  at  their  intersec-  PIG<  59. 

tion.      An    estimated    dis- 
tance may  be  often  verified  by  a  cast  of  the  lead  where  sound- 
ings vary  considerably. 

Suppose  in  Fig.  59,  AB  =  h  =  the  height  in  feet  of  an 
object  whose  angular  altitude  at  C=  a  .     Then  if  ABC  =  90° 

and  BC  —  d,  d  =•  • .     Now  as  a  is  very  small  and  ex- 
tan  a 

pressed   in   minutes    of   arc,   tan  a  =  a   sin    1';    therefore, 

d  = - ? ,  and  to  express  d  in  nautical  miles,  divide  sec- 

a  sin  V 
ond  member  by  6080.27;  substituting  value  of  sin  V  we  have 

d  (in  sea  miles) 

_  h  (in  feet) h^          1        __  557  _^ 

~  a  (6080.27)  X  .00029  ="  a  '  1.76328  "  a  ' 

Ex.  21. — A  lighthouse  140  feet  high  subtends  an  angle  of 

16';  find  the  distance  in  nautical  miles. 

140 
d  =  -jg    X  .567  =  4.96  nautical  miles. 

*  The  Barr  and  Stroud  range-finder  is  sufficiently  accurate  for  navigational  pur- 
poses at  distances  varying  between  800  and  7000  yards,  and  its  use  is  practicable 
on  board  ship. 


210 


NAVIGATION 


Table  33  of  Bowditch  gives  the  distance,  by  vertical  angle, 
at  intervals  of  one-tenth  of  a  mile  up  to  5  sea  miles  for  ob- 
jects whose  heights  vary  from  40  to  2000  feet. 

In  this  method  B,  the  foot  of  the  object,  should  be  seen  and 
the  angle  ABC  should  be  90° ;  for  this  reason  the  observer's 
eye  should  be  as  low  down  as  possible.  The  error  due,  how- 
ever, to  a  slight  height  of  the  eye  is  inappreciable,  but  that 
due  to  the  visible  shore  line  B',  not  being  at  B  the  foot  of  the 
object,  might  be  material.  In  other  words,  B'  should  be  at  B 
and  C"  at  C. 

A 


Dip 


D 

FIG.  60. 


Buckner's  method  is  one  often  used  for  finding  the  distance 
of  a  target,  and  may  be  used  for  finding  the  distance  of  an 
isolated  object  beyond  which  the  sea  horizon  can  be  seen.  In 
Fig.  60,  AB  is  the  height  of  eye  above  sea-level;  EAC,  the 
dip  of  sea  horizon  C  due  to  height  h  =  AB;  D,  isolated  object ; 
a,  the  angle  between  the  object  and  sea  horizon  beyond;  and 

h  (in  feet) 
a  (in  yards)  = 


3  tan  (a-j-Dip) 

Table  34  of  Bowditch  gives  the  distance  in  yards  for  various 
values  of  angle  a  observed  at  heights  of  observer's  eye  varying 
from  20  to  120  feet.  Lecky's  off-shore  distance  tables  are  more 
extended  in  their  application,  likewise  Von  Bayer's  diagram. 

To  find  the  distance  of  an  object  of  known  height,  just  vis- 
ible on  the  sea  horizon  to  an  observer's  eye  of  a  given  height, 


Fix  BY  CHANGE  OF  BEARING  211 

it  is  sometimes  useful  to  use  Table  6  of  Bowditch  which  gives 
the  distance  of  visibility  of  objects  at  sea  for  different  heights ; 
the  distance,  owing  to  the  uniform  curvature  of  the  sea,  de- 
pending on  the  heights  both  of  object  and  observer's  eye. 

Ex.  22. — From  a  height  of  45  feet,  a  light,  whose  height 
from  the  light  list  is  160  feet,  is  seen  to  disappear  below  the 
horizon ;  find  its  distance  in  nautical  miles. 

For    45  feet,  distance    7.7  miles. 

For  160  feet,  distance  14.5  miles. 


Kequired  distance  =  22.2  miles. 

Table  6  of  Bowditch  is  calculated  from  the  formula  (to  be 
deduced  later  on) 

d  =  1.148  V  K 

li  =  height  of  object  in  feet,  d  =  distance  in  nautical  miles 
of  the  object  just  visible  in  the  horizon,  the  observer's  eye 
being  at  the  surface  of  the  earth. 

Now,  if  li'  =  the  height  of  eye  in  feet, 
d  =1.148  (V^+  V^')- 

Ex.  23. — An  observer,  height  of  eye  36  feet,  sees  in  the 
sea  horizon  the  top  of  a  lighthouse  known  to  be  121  feet  high. 
What  is  the  distance  in  nautical  miles  ? 
d  =  1.148  (V36  +  V121)  =  1-148  X  17  =  19.55  miles. 

117.  (C)  By  two  bearings  of  a  single  object  and  the 
course  and  distance  run  in  the  interval. — (1)  This  involves 
the  solution  of  a  plane  triangle,  given  one  side  and  the  two 
adjacent  angles.  The  ship  is  steering  the  course  AB  (Fig. 
61).  At  first  observation,  ship  is  on  the  line  of  bearing  AC 
and  the  patent  log  is  read.  At  second  bearing,  ship  is  on  the 
line  of  bearing  BC  and  patent  log  is  again  read.  The  dif- 
ference of  readings  of  patent  log  gives  the  distance  AB, 
from  the  course  and  bearings  the  angles  A  and  B  are  known, 
and  C  =  180—  (B  +  A). 


212 


NAVIGATION 


FIG.  61. 


(1)  By  plane  trigonometry. — 

. ,-.       sin  B  .      A  -r>       •,  •pj.o,       sin  A.  . 
AG  =  imtf  X  AB  and  S0  =  IhH7  X 
The  distance  of  passing  abeam  CD  =  BC  sin  B. 

Tables  have  been  compiled  giving  factors  by  inspection; 
the  factor  of  the  first  column  multiplied  by  distance  run  be- 
tween bearing  lines  gives  the  distance 
of  object  at  second  bearing,  the  factor 
of  the  second  column  multiplied  by  the 
distance  run  gives  the  distance  when 
the  object  was  abeam.  The  arguments 
in  these  tables  are  the  difference  be- 
tween course  and  first  bearing,  differ- 
ence between  course  and  second  bear- 
ing ;  the  difference  of  bearings  in  Table 
5 A,  Bowditch,  being  at  intervals  of 
quarter  points,  in  Table  5B  at  intervals 

of  two  degrees;  the  factors  in  each  table  being  for  a  distance 
run  of  one  mile. 

So  far  as  the  factors  are  concerned,  it  is  immaterial  whether 
the  course  and  bearings  are  per  compass,  magnetic,  or  true, 
provided  they  are  all  from  the  same  meridian. 

Ex.  24-— A  ship  heading  292°  (p.  c.),  var.  +  12°,  dev. 
-(-9°,  had  a  lighthouse  bearing  234°  (p.  c.)  ;  after  a  run  of 
10  miles,  the  same  light  bore  166°  (p.  c.). 

Find  the -distance  at  second  bearing,  also  when  light  was 
abeam. 

Difference  between  course  and  first  bearing  58°. 
Difference  between  course  and  second  bearing  126°. 
Table  5B,  factor  from  first  column  =  0.91  and  distance  at 
second  bearing  =  9.1  miles. 

Table  5B,  factor  from  second  column  =  0.74  and  distance 
abeam  =  7.4  miles. 

It  is  to  be  understood  that  the  course  and  distance  are 


DOUBLING  ANGLE  ON  THE  Bow  213 

those  over  the  ground,  and  due  allowance  must  be  made  for 
currents. 

A  simple  application  of  this  method  is  what  is  known  as 
the  "  bow  and  beam  bearing."  The  bearing  of  an  object  is 
taken  when  it  is  45°  on  the  bow,  and  patent  log  noted;  an- 
other bearing  is  taken  when  the  object  is  abeam  and  patent 
log  again  read.  The  distance  run  between  the  times  of  the 
two  bearings  is  the  distance  of  the  object  when  abeam,  pro- 
vided the  course  and  distance  have  not  been  influenced  by 
currents  or  bad  steerage. 

If  the  first  bearing  was  taken  abeam  and  the  second  when 
the  object  was  on  the  quarter,  the  distance  run  multiplied  by 
1.4  will  give  the  distance  of  the  object  at  second  bearing,  the 
distance  run  being  the  distance  of  object  when  abeam. 

(2)  The  graphic  solution  of  this  method  is  easier  and,  cer- 
tainly, more  general  than  the  factor  solution. 

By  means  of  parallel  rulers,  draw  the  first  line  of  bearing 
AC  (Fig.  61)  on  the  chart.  When  the  bearing  has  changed 
sufficiently  draw  the  second  line  of  bearing  on  the  chart. 
Cut  these  lines  by  one  representing  the  course,  transferred 
from  the  compass  rose  by  parallel  rulers,  so  that  the  distance 
intercepted  between  the  two  lines  of  bearing  shall  equal  the 
distance  run.  If  the  course  used  was  the  course  made  good, 
and  no  current  affected  the  run,  the  points  of  intersection  of 
the  course  line  with  the  lines  of  bearing  will  be  the  positions 
of  the  ship  at  the  times  of  taking  the  bearings. 

(3)  Doubling  the  angle  on  the  bow. — In  this  case  note 
the  patent  log  when  the  object  has  a  certain  bearing  on  the 
bow,  or,  in  other  words,  when  there  is  a  certain  angle  between 
the  ship's  head  and  the  object;  again  note  the  patent  log 
when  this  bearing  on  the  bow,  or  angle,  has  doubled ;  the  dif- 
ference between  the  two  readings  of  the  patent  log,  or  the 
distance  run,  is  the  distance  of  the  object  at  the  second  bear- 
ing. 


214 


NAVIGATION 


This  is  shown  graphically  in  Fig.  62.  Since  DBG  =  WAO 
and  also  equals  DAG  +  BCA,  BOA  =  DAC  and  BC  =  AB 
—  distance,  run. 

Hence,  the  rule  "  when  the  angle  on  the  bow  is  doubled, 
the  distance  of  the  object  at  the  second  bearing  equals  the 
distance  run  " ;  provided  there  is  no  current. 

A  special  application  of  this  method,  and  one  universally 
used,  is  the  four-point  bearing,  the  double  angle  being  90°, 


FIG.  62. 


FIG.  63. 


and  the  distance  run  the  distance  when  abeam.  A  com- 
mon application  is  when  the  first  bearing  is  two  points 
on  the  bow,  giving  the  distance  on  the  bow  as  distance  run; 
again  when  the  first  bearing  is  60°  on  the  bow,  the  distance 
run  will  be  the  distance  of  object  when  it  bears  30°  abaft  the 
beam. 

Distance  of  passing  an  object  abeam. — In  case  the  first 
angle  on  the  bow  is  26£°  and  the  second  angle  is  45°,  the 
distance  run  between  the  two  bearings  will  be  the  distance 
of  passing  the  object  abeam,  if  course  and  distance  are  un- 
affected by  current.  A  knowledge  of  this  distance  is  of  im- 
portance as  the  point  is  approached. 


DANGER  ANGLE  215 

118.  Danger  angle. — When  sailing  along  a  coast  and  it  is 
desired  to  avoid  sunken  rocks,  or  shoals,  or  dangerous  obstruc- 
tions at  or  below  the  surface  of  the  water,  and  which  are 
marked  on,  the  chart,  the  navigator  may  pass  these  at  any  de- 
sired distance  by  using  what  is  known  as  a  danger  angle,  of 
which  there  are  two  kinds,  a  horizontal  and  a  vertical  danger 
angle;  the  former  requires  two  well-marked  objects  projected 
on  the  chart,  lying  in  the  direction  of  the  coast,  and  suffi- 
ciently distant  from  each  other  to  give  a  fair-sized  horizontal 
angle;  the  latter  requires  a  well-charted  object  of  known 
height. 

In  Fig.  63,  let  AMB  be  a  portion  of  a  coast  along  which  a 
vessel  is  sailing  on  course  CDf  A  and  B  two  prominent  objects 
projected  on  the  chart;  S  and  S'  are  two  outlying  shoals, 
reefs,  or  dangers. 

To  pass  at  a  given  distance  outside  the  center  of  danger  8'. 
— With  the  middle  point  of  danger  as  a  center,  and  the  given 
distance  as  a  radius,  describe  a  circle;  pass  a  circle  through 
A  and  B  tangent  to  the  seaward  side  of  the  first  circle.  To 
do  this  practically,  it  is  only  necessary  to  join  A  and  B,  and 
draw  a  line  perpendicular  to  the  center  of  AB,  then  ascertain 
by  trial  the  location  of  the  center  of  circle  EAB.  Measure 
the  angle  a  =  AEBf  set  sextant  to  this  angle,  and  remember- 
ing that  AB  subtends  the  same  angle  at  all  points  of  the  arc 
AEB,  the  ship  will  be  outside  the  arc  AEB,  and  clear  of 
danger  $',  as  long  as  AB  does  not  subtend  an  angle  greater 
than  a  to  which  the  sextant  is  set. 

Now,  should  it  be  desired  to  pass  a  certain  distance  inside 
of  a  danger  S — with  the  middle  point  of  danger  as  a  center 
and  the  desired  distance  as  a  radius  describe  a  circle;  pass  a 
second  circle  through  A  and  B  tangent  to  this  circle  at  G. 
Measure  the  angle  BGA  =  <£  with  a  protractor.  Then,  as 
long  as  the  chord  AB  subtends  an  angle  greater  than  <£,  the 
ship  will  be  inside  the  circle  A  GB  and  clear  of  danger  8. 


216 


NAVIGATION 


Should  both,  dangers  exist  and  it  be  desired  to  pass  between 
them  with  a  margin  of  safety  already  referred  to,  steer  on 
course  CD  so  that  the  angle  subtended  by  AB  shall  be  <  a 
but  >  </>. 

119.  Vertical  danger  angle. — Practically  the  same  general 
principle  is  involved  in  this  as  in  the  horizontal  danger  angle ; 
however,  only  one  object  is  used  and  that  one  must  be  of 
known  height. 


FIG.  64. 


In  Fig.  64,  draw  circles  around  the  dangers  8  and  8'  with 
radii  representing  a  safe  margin  of  safety.  Let  A B  be  an 
object  of  known  height.  With  A  as  a  center  draw  circles 
tangent  at  E  and  G.  Measure  the  distances  AE  and  AG;  find 
from  Table  33,  Bowditch,  or  by  computation,  the  angular 
altitude  of  AB  of  known  height  for  distance  AE,  let  it  be  a; 
also  for  distance  AG,  let  it  be  <j>.  To  pass  outside  of  and 
clear  of  8'  the  angular  height  of  A  B  must  be  <  a.  To  pass 
inside  and  clear  of  8  the  angular  altitude  of  AB  must  be  >  <j>. 
These  are  the  limits  for  ensuring  a  safe  passage  between  8 


DANGER  BEARING 


217 


and  '8'.  In  observing  a  vertical  danger  angle,  the  observer 
should  be  as  near  the  water  line  as  possible  to  minimize  errors 
due  to  height  of  eye,  and  the  angle  at  A  should  be  90°. 

120.  Danger  bearing. — A  bearing,  properly  taken  and  used, 
may  often  be  of  great  use  in  keeping  a  ship  out  of  danger. 
Suppose  C  and  C"  to  be  shoals  or  rocks  near  the  coast  (Fig. 
65).  A  ship  is  passing  on  course  efg.  Lay  down  on  the 
chart  the  tangents  AD  and  Ag  with  any  desired  margin  of 
safety.  With  A  as  a  center  describe  the  arcs  of  circles  to  in- 
clude dangers  C  and  C'.  Note  the 
magnetic  bearing  DA  and  gA  and 
find  what  should  be  the  compass 
bearings  for  the  given  course.  Now 
before  the  distance  of  A  is  reduced 
to  the  radius  of  the  circle's  arc  en- 
closing C,  the  bearing  of  A  must  be 
to  the  left  of  the  danger  bearing 
DA  and  kept  so  to  avoid  danger. 
The  bearing  of  A  must  not  get  to 
the  left  of  gA  till  the  distance  of  A 
is  greater  than  the  radius  of  the  arc 
enclosing  C".  It  may  often  be  pos- 
sible to  find  a  danger  bearing  on  a 
range;  for  instance,  if  A  and  B  are 
in  range  on  the  danger  bearing  !M,-the  object  B  must  be 
kept  open  to  the  right  to  ensure  safety,  and  the  guarantee  is 
better  than  a  compass  bearing  would  give. 

Lights  as  danger  guides. — Lights  of  lighthouses  may  be 
used  to  give  warning  of  danger  as  the  object  A  was  used  above. 
Many  lights,  showing  white  over  safe  waters,  show  red  over 
sectors  embracing  areas  of  rocks,  shoals,  or  depths  over  which 
the  approaches  would  be  dangerous,  and  it  is  the  duty  of  the 
navigator  to  keep  out  of  the  danger  sectors.  The  magnetic 
bearings  showing  the  limits  of  the  sectors  are  given  from  sea- 


FIG.  65. 


218  NAVIGATION 

ward,  and  the  bearing  of  the  light,  taken  frequently,  should 
not  be  allowed  tu  get  within  the  danger  sector  unless  the  dis- 
tance of  the  light  is  known  to  be  greater  than  the  radius 
enclosing  the  dangers. 

A  navigator  is  furnished  with  charts,  light  lists,  and  sailing 
directions,  all  of  which  give  details  as  to  color,  character, 
and  visibility  of  navigational  lights,  and  the  navy  regulations 
make  it  his  duty  to  become  thoroughly  conversant  with  these 
details  before  coming  within  their  range  of  visibility. 

It  may  sometimes,  in  fact,  has  often  happened,  that  one  light 
is  sighted  when  the  run  indicates  the  ship  to  be  in  the  region 
of  another  light  for  which  a  lookout  has  been  kept;  hence, 
the  rule,  on  sighting  a  light,  is  to  compare  its  visible  character- 
istics with  those  laid  down  in  the  light  lists,  and,  if  appar- 
ently not  a  fixed  light,  the  duration  of  its  periods  must  be 
noted  by  watch. 

It  must  not  be  forgotten,  however,  that  abnormal  atmos- 
pheric conditions  may  increase  the  range  of  visibility  of  a 
light,  whilst  mist  may  decrease  it  and  is  often  found  to  make 
white  lights  appear  red.  When  a  fixed  light  is  first  sighted, 
especially  under  fair  conditions,  and  there  is  any  question  as 
to  whether  it  is  a  lighthouse  or  a  vessel's  light,  simply  descend 
a  short  distance,  and  again  look  for  the  light.  A  vessel's  light 
is  of  limited  intensity  and,  if  seen  at  all,  can  be  seen  at  any 
height;  a  navigation  light  can  be  seen,  as  a  rule,  as  far  as 
the  curvature  of  the  earth  will  permit  and  this  distance,  de- 
pending on  the  height  of  eye  and  of  the  light,  will  be  lessened 
by  the  observer  going  lower  down.  A  descent  of  a  few  feet 
may  cause  a  navigation  light  to  disappear. 

121.  Fog  signals. — A  navigator  should  make  himself  famil- 
iar with  all  fog  signals  of  the  locality  in  which  he  is  cruising; 
in  foggy  weather  and  in  the  neighborhood  of  signals,  the 
closest  attention  must  be  paid  in  an  effort  to  hear  them  and 
locate  their  direction.  When  heard,  their  periods  should  be 


FOG  SIGNALS  219 

timed  and  a  comparison  made  with  those  recorded  in  the  light 
lists  to  ensure  identification.  However,  it  must  not  be  for- 
gotten that  atmospheric  conditions  affect  the  transmission  of 
sound  and  at  times  cut  it  off  entirely,  producing  a  silent  zone 
in  a  locality  where  the  signal  could  be  heard  distinctly  at 
other  times;  hence,  if  dependent  on  aerial  signals,  in  a  fog, 
slow  the  ship,  navigate  with  extreme  caution,  keep  the  lead 
going  on  soundings,  make  every  effort  to  guard  against  over- 
logging,  and,  as  a  last  resort,  if  the  depth  of  water  will  permit, 
anchor  the  ship.  Ordinarily  the  sound  of  an  aerial  fog  sig- 
nal, if  it  is  to  leeward,  will  be  heard  sooner  from  aloft;  if  to 
windward,  from  a  point  nearer  the  water. 

Fortunately,  the  cases  are  now  rare  in  which  the  navigator 
has  to  depend  on  aerial  signals,  for  the  submarine-bell  has 
been  generally  adopted  by  maritime  nations  as  a  means  of 
making  fog  signals  (see  Appendix  A). 

In  using  this  system,  listen  with  the  starboard  receiver  when  the  bell  is  known  to 
be  on  the  starboard  side,  otherwise  use  the  port  receiver;  at  all  events  if  the  bell 
is  heard  in  both  receivers  its  direction  will  be  shown  as  being  to  starboard  or  port 
by  the  greater  intensity  of  sound  in  the  starboard  or  port  receiver,  and  the  correct 
bearing  of  the  bell  may  be  obtained  by  so  changing  the  course  as  to  bring  it  directly 
ahead,  at  which  position  the  intensity  of  sound  becomes  the  same  in  both  receivers, 
provided  the  listener  can  hear  equally  well  in  both  ears;  otherwise  there  will  be  an 
error  of  direction,  the  amount  being  dependent  on  the  difference  of  hearing  in  the 
listener's  ears. 

The  submarine  signal  is  more  reliable  than  the  aerial  signal  since  it  can  be  heard 
on  board  vessels  fitted  with  receivers  at  greater  distances,  these  distances  varying 
according  to  condition  of  instruments,  attention,  and  delicacy  of  hearing  of  listener; 
its  direction  can  be  ascertained  with  fair  accuracy,  and  the  sound  is  not  subject  to 
the  silent  zone,  though  there  is  always  a  possibility  that  the  bell's  mechanism  may 
be  deranged. 

An  approximate  fix  may  be  obtained  from  the  bearing  of  the  bell  combined  with 
a  sounding  or  with  a  line  of  position  (if  recent  and  reliable)  brought  up  to  the 
instant  of  locating  the  direction  of  the  bell;  also  from  two  bearings  of  the  bell  and 
the  course  and  distance  run  in  the  interval. 

However,  in  a  fog,  even  when  within  hearing  of  a  sub- 
marine-bell whose  direction  may  be  fairly  well  determined, 
the  navigator  must  be  watchful  and  cautious,  should  reduce 
speed  and  make  a  judicious  use  of  both  log  and  lead,  bearing 
in  mind  the  fact  that  the  effect  of  cross  currents  encountered 
will  be  increased  as  speed  is  reduced. 


CHAPTER  VI. 
THE  SAILINGS. 

122.  The  position  of  a  ship  at  sea,  at  a  given  moment,  is 
denned  by  its  latitude  and  longitude.  This  position  is  con- 
nected with  one  left,  or  with  one  to  which  the  vessel  is  bound, 
by  the  true  course  and  distance  between  them. 

A  course  and  distance  can  be  resolved  into  difference  of 
latitude  and  departure,  and  this  departure  converted  into 
difference  of  longitude;  so  that  knowing  the  course  and  dis- 
tance sailed  from  a  given  position,  the  latitude  and  longitude 
of  the  position  arrived  at  can  be  found ;  or,  when  desired,  the 
course  and  distance  between  two  given  positions  may  be  found. 
The  various  methods  of  solution  of  these  problems  are  termed 
Sailings,  and  include,  Plane,  Parallel,  Middle  Latitude,  and 
Mercator  Sailings. 

The  term  "  Dead  Reckoning  "  includes  all  calculations  to 
determine  a  ship's  position,  given  only  the  true  courses  and 
distances  run  from  a  given  point  of  departure.  It  involves 
the  principles  of  the  various  sailings  explained  below. 

The  latitude  and  longitude  determined  by  "  Dead  Reckon- 
ing" are  noted  thus,  "Lat.  by  D.  R.,"  "Long,  by  D.  R." 

The  position  by  D.  R.  is  liable  to  error  due  to  bad  steering, 
improper  logging  of  the  distances  run,  faulty  allowance  for 
leeway,  effects  of  wind,  currents,  etc.,  and  the  exact  position 
of  the  ship,  out  of  sight  of  land,  can  be  determined  only  by 
celestial  observations. 

The  term  "  Day's  Work,"  though  frequently  applied  so  as 
to  embody  only  dead  reckoning,  the  finding  of  the  0  and  d 
made  by  D.  R.  from  the  point  of  departure,  and  the  C  and  d  by 
D.  R.  from  position  arrived  at  to  destination,  should  properly 
include  results  arising  from  a  knowledge  of  the  ship's  true 


TAKING  THE  DEPARTURE  221 

position  obtained  either  by  bearings  or  celestial  observations 
as  indicated  in  Art.  310. 

Particular  attention  must  be  paid  to  plane  and  parallel 
sailings  as  a  full  understanding  of  their  principles  is  essential 
to  an  understanding  of  middle  latitude  sailing  which  is  gen- 
erally employed  for  short  distances,  as  in  a  day's  run;  for 
longer  distances  it  will  be  better  to  use  Mercator  sailing,  which 
is  a  method  based  on  the  principles  already  explained  in  the 
articles  on  the  Mercator  chart. 

The  methods  of  laying  down  a  ship's  run  and  finding  the 
position  graphically  have  been  explained  in  Chapter  II. 

123.  Taking  the  departure. — On  leaving  port,  at  the  begin- 
ning of  a  voyage,  the  ship's  position  is  fixed  by  some  one  of  the 
methods  explained  in  the  chapter  on  "  Fixing  positions  near 
land,"  the  best  method  available  at  the  time,  of  course,  being 
used;  and  from  a  last  position  thus  obtained,  the  succeeding 
traverse,  as  laid  down  graphically  on  the  chart,  takes  its  com- 
mencement. This  final  position  may  be  taken  from  the  chart 
and  be  considered  the  point  of  departure  from  which  future 
positions  may  be  calculated  in  the  navigator's  work  book. 
However,  it  is  frequently  the  custom  to  take  from  the  last 
position  at  which  objects  can  be  distinctly  seen,  the  bearing 
and  distance  of  some  fixed  point  of  land,  lighthouse,  light 
vessel,  or  beacon,  whose  latitude  and  longitude  are  known,  and 
to  consider  its  reversed  true  bearing  and  distance  as  a  true 
course  and  distance  sailed  by  the  vessel,  thus  taking  its  posi- 
tion as  the  point  of  departure.  If  the  distance  is  not  known, 
it  must  be  estimated.  This  is  what  is  known  as  taking  a 
departure. 

The  navigator  must  be  careful  to  correct  the  compass  bear- 
ing of  this  point  for  the  variation,  and  the  deviation  of  the 
compass  due  to  the  ship's  heading  at  the  time  of  taking  the 
departure,  then  to  reverse  the  true  bearing  to  get  the  true 
course  the  ship  is  assumed  to  have  sailed.  This  reversed  true 
bearing  is  called  the  departure  course,  and  it  equals  the  true 
bearing  ±180°. 


222  NAVIGATION 

When  a  departure  is  thus  taken  the  departure  course  and 
distance  appear  in  the  record  of  the  first  day's  work,  in  the 
proper  columns  of  the  tabulated  form,  and  are  treated  like 
any  other  course  and  distance. 

Noon  position  as  a  point  of  departure. — In  the  succeeding 
part  of  the  voyage,  each  noon  position  by  observation  is  taken 
as  a  new  point  of  departure. 

Latitude  left  and  longitude  left. — In  dead  reckoning,  these 
terms  refer  to  the  latitude  and  longitude  of  the  point  of 
departure. 

Latitude  in  and  longitude  in. — These  terms  in  dead  reck- 
oning apply  to  the  latitude  and  longitude  arrived  at,  and  are 
marked  "  by  D.  K."  when  by  account,  or  "  by  obs."  when  by 
observation. 

Course  and  distance  made  good. — For  a  given  interval  of 
time  the  course  and  distance  from  the  last  point  of  departure 
to  the  position  by  observation  at  the  end  of  that  time  are  the 
course  and  distance  made  good. 

124.  The  following  notation  will  be  followed  in  this  book 
whenever  it  may  be  necessary  to  represent  the  quantities  re- 
ferred to  below: 

C  will  represent  the  course  measured  from  the  North 
or  South  towards  East  or  West. 

CN     "         "  "    course     measured     from     North 

around  to  the  right  from  0°  to 
360°. 

L     "         "  "   latitude. 

A      "          "  "    longitude. 

L!    "         "  "    latitude  of  place  left. 

A!     "         "  "    longitude  of  place  left. 

L2    "  "   latitude  of  place  arrived  at. 

A2     "  "   longitude  of  place  arrived  at. 

l=L2~L±  "         "  "    difference  of  latitude. 

D—X^^  "         "  "    difference  of  longitude. 

p      "  "    departure. 

d      "  "    distance  sailed  on  course  C. 

LQ    "         "  "   the  middle  latitude  = 


PLANE  SAILING 


223 


Latitude  is  North,  or  South  according  as  the  place  is  in  the 
Northern  or  Southern  hemisphere.  Longitude  is  East  or 
West  according  as  the  place  is  on  a  meridian  East  or  West  of 
Greenwich. 

In  the  solutions  by  computation  of  the  triangles  of  plane, 
middle  latitude  and  Mercator  sailings,  the  course  C  is  an 
interior  angle  of  the  triangle  and  is  not  greater  than  90°. 
Its  general  direction  is  determined  by  I  and  p  or  m  and  D. 
Having  been  found,  C  should  be  expressed  as  (7N  for  practical 
purposes  (Exs.  39,  52,  and  53). 

If  the  data  includes  the  course  as  (7N,  express  it  as  C  and 
indicate  its  proper  direction  before  proceeding  with  the  com- 
putation (Ex.  38). 

In  solutions  by  inspection,  as  in  dead  reckoning,  the  course 
should  be  retained  in  the  form  of  (7N  as  the  traverse  tables  are 
tabulated  for  courses  up. to  360°  (Ex.  25). 
PLANE  SAILING. 

125.  For  small  distances  at  sea,  the  curvature  of  the  earth 
may  be  neglected,  and  the  small  portion  of  the  earth  passed 
over  may  be  regarded  as  a  plane  surface, 
on  which  the  meridians  are  parallel  right 
lines  perpendicular  to   the   equator,   the 
parallels  of  latitude  are  right  lines  paral- 
lel to  the  equator,  and  the  length  of  a  de- 
gree is  assumed  the  same  whether  meas- 
ured on  the  equator,  meridian,  or  parallel. 

Though  this  assumption  is  not  strictly 
correct,  the  results  obtained  by  plane  sail- 
ing may  be  considered  sufficiently  exact  FIG.  66. 
for  any  ordinary  day's  run. 

The  relations  existing  between  the  parts  that  enter  into 
plane  sailing  are  indicated  in  a  right  triangle  in  which  C 
is  the  course,  I  the  difference  of  latitude,  p  the  departure  in 
the  latitude  left  or  that  arrived  at.  From  an  application  of 


224 


NAVIGATION 


the  principles  of  plane  trigonometry,  the  relations  are  from 
Fig  66, 

Z  =  d  cos  ai 

p  =  dsmC,[  (115) 

p  =  I  tan  0.  j 

The  solution  of  the  above  equations  is  facilitated  by  the  use 
of  Tables  1  and  2,  Bowditch  (which  are  tables  for  the  solu- 
tion of  any  right  triangle),  calling  d  the  hypothenuse,  Z  the 
side  adjacent,  and  p  the  side  opposite  the  course  C. 

Table  1  gives  the  courses  in  quarter  points  and  distances 
up  to  300  for  each  unit;  Table  2  gives  courses  in  degrees  and 
distances  to  600. 

Should  the  distance  for  which  I  and  p  are  desired  be  greater 
than  the  limit  of  the  table,  subdivide  the  distance  into  two  or 
more  parts,  finding  the  I  and  p  for  these  separate  parts  and 
adding;  thus  the  cliff,  of  lat.  for  1340  miles,  course  10°,  will 
be  the  diff.  of  lat.  for  600  +  cliff,  of  kt.  for  600  +  diff.  of 
lat.  for  140,  course  10°,  and  the  departure  may  be  found  in 

the  same  way;  or  1340  may  be  di- 
vided by  4,  giving  335,  then  I  and 
p  may  be  found  for  335,  course 
10°,  and  multiplied  by  4;  simi- 
larly any  other  factor,  as  5  or  10, 
might  be  used. 

Since  I  =  d  cos  0  and  p  =  d 
sin  C,  it  is  apparent  that  the  I 
and  p  for  any  course  are  re- 
spectively the  p  and  I  for  the 
complement  of  the  course,  as 
shown  in  the  tabulated  form. 

So  it  is  apparent  that  all  ques- 
tions involving  C,  d,  I,  and  p  can 
be  solved  by  inspection  by  using 
Any  expression  involving  sines, 


Table  2. 


Diif  .  of  Lat.  and  dep.  for 

10°  (170°,  190°,  350°) 

Dist. 

Lat. 

Dep. 

I 

1.0 

0.2 

2 

2.0 

0.3 

3 

3.0 

0.5 

4 

3.9 

0.7 

5 

4.9 

0.9 

6 

5.9 

1.0 

7 

6.9 

1.2 

8 

7.9 

1.4 

9 

8.9 

1.6 

10 

9.8 

1.7 

Dist. 

Dep. 

Lat. 

80°  (100°,  *803,  280°) 

Tables  1  and  2,  Bowditch. 


PLANE  SAILING  225 

cosines,  secants,  tangents,  or  cotangents  of  the  following  forms 
may  be  referred  to  the  traverse  table. 

Thus  x  —  30  sin  60°,  ]          Also  35  =  x  tan  50°, 
If  p=d  sin  60°,          L          If  p  = /tan  50°, 
When  d  =  30,  find  p.  J          When  p  —  35,  find  L 

A  triangle  may  be  solved  by  the  "Rule  of  Sines"  in  the 
same  way. 

When  the  distance  sailed  is  so  great  that  the  curvature  of 
the  earth  cannot  be  neglected. — In  Fig.  67,  let  P  be  the  ele- 
vated pole  of  the  earth.  0  and  A  two  places  on  the  surface 
connected  by  the  loxodrome  CA.  Let  C'A!  be  an  arc  of  the 
equator  intercepted  between  the  meridians  of  G  and  A. 

Consider  the  distance  CA  to  be 
divided  into  a  very  large  number  of 
equal  distances;  each  distance  form- 
ing with  its  corresponding  differ- 
ence of  latitude  and  departure  a 
right  triangle.  All  these  triangles 
are  similar,  two  angles  of  each  tri- 
angle, the  right  angle  and  the  course 
C,  being  equal  to  the  corresponding 
angles  in  the  other  triangles.  Each 
triangle  is  so  small  that  it  may  be 

taken  as  a  plane  right  triangle.     Such  would  be  the  triangle 
dbc  for  the  small  distance  ca. 

Now  letting  d±,  d2 ,  ds . . . .  dn  be  the  small  distances  into 

which  CA  is  divided; 

Z± ,  12 ,  ls ln,  their  corresponding  differences  of  latitude 

for  the  course  C; 

Pi,  p2,  p3 pn,  the  corresponding  departures  for  the  same 

course,  each  departure  being  measured  in 
the  latitude  of  its  own  triangle ; 


226  NAVIGATION 

we  have 

?j_   —  di  cos  C,  12   =  d2  cos  C ...  .ln   —  dn  cos  (7; 
p±  =:  t?±  sin  (7,  p2  =  tZ2  sin  (7 . . . .  pn  =  dn  sin  (7. 
Therefore, 

k   +  Z2   +  Zs  •  -  -  -  k '  =  K  +  4  +  d3 . . . .  dn)  cos  (7. 

Fi  +  P2  +  ^s ^n  =  (di  +  d2  +  ds dn)  sin  C. 

Now  since  the  parallels  of  latitude  through  (7  and  A  are 
the  same  distance  apart  on  all  meridians,  the  difference  of 
latitude  of  C  and  A  is  the  sum  of  the  partial  differences 
of  latitude,  and,  as  the  total  distance  is  the  sum  of  the  partial 
distances,  we  have 

1  =  k  +  h  +  k  -  -  -  *»  =  K  +  d2  +  d, . . .  dn)  cos  C, ) 

or  I  =  d  cos  C.  \  ( 

And  if  each  partial  departure  is  measured  in  the  latitude  of 
its  own  triangle,  the  sum  of  these  partial  departures  will 
represent  the  true  departure  in  the  triangle  CAB,  and  hence, 
P  =  Pi  +  p2  +  Ps'-Pn=(di+d2  +  d3..  dn)  sin  C, ) 
or  p  =.  d  sin  C.  ) 

So  that  I  and  p  are  calculated  by  the  same  formulae  whether 
the  curvature  of  the  earth  is,  or  is  not,  considered.  However, 
the  sum  of  the  partial  departures  is  less  than  the  distance  be- 
tween the  meridian  left  and  the  meridian  arrived  at  measured 
in  the  lower  latitude,  and  greater  than  that  measured  in  the 
higher  latitude,  and  is  approximately  equal  to  the  departure 
of  the  parallel  midway  between  the  two. 

When  both  points  of  departure  and  arrival  are  on  the  same 
side  of  the  equator  the  latitude  of  the  parallel  midway  between 
is  known  as  the  middle  latitude,  and  is  equal  to  the  half  sum 

of  the  two  latitudes;  in  other  words,  L0  =  -  x  ~j~ — 2.       (118) 

From  what  has  been  said  above,  it  is  evident  that  a  ship 
sailing  due  North  or  South  (true)  remains  on  the  meridian, 
changes  her  latitude  only,  and  the  distance  sailed  is  simply  a 


TRAVERSE  SAILING  227 

difference  of  latitude,  is  either  "  Northing  "  or  "  Southing/' 
and  must  be  so  entered  in  the  tabulated  form  of  work.  When 
a  ship  sails  due  East  or  West  (true),  she  remains  on  her  paral- 
lel, does  not  change  her  latitude,  and  the  distance  sailed  is 
either  "  Easting  "  or  "  Westing,"  and  must  be  so  entered  in 
the  form  for  work,  this  departure  to  be  later  converted  into  its 
proper  difference  of  longitude. 

When  a  ship  sails  due  East  or  West  (true),  on  the  equator, 
the  distance  East  or  West  is  itself  difference  of  longitude. 

When  a  ship  sails  on  a  loxodrome,  at  an  acute  angle  with  the 
meridian,  she  alters  both  her  latitude  and  longitude. 

TRAVERSE    SAILING. 

126.  If  a  ship  sails  on  several  courses  instead  of  a  single 
course,  she  makes  an  irregular  track,  called  a  traverse,  and  it 
is  the  function  of  traverse  sailing  to  find  the  single  course 
and  distance  that'  would  have  taken  the  ship  to  the  position 
arrived  at,  in  other  words,  the  resultant  course  and  distance 
as  well  as  the  corresponding  difference  of  latitude  and  de- 
parture. 

If  C± ,  C2  .  .  .  .  Cn  be  the  different  courses,  and 
d± ,  d2  .  .  .  .  dn  the  corresponding  distances, 
then,  Z±  =  d±  cos  Clfl2  =  d2  cos  C2  ....  ln  =  dn  cos  Cn; 
Pi  =  di  sin  C19  £>2  —  d2  sin  (72  ....  pn  =  dn  sin  Cn; 
and,  as  before,  I  =  Zx  +  12  +  13  .  -  .  .  In , 

P  =  P±  +  P2  +  Ps  •  •  •  -  Pn  , 
p  being  measured  along  the  parallel .   1  ~^~ 2,  or,  as  in  the  case 

a 

of  a  single  course,  "the  whole  difference  of  latitude  is  equal 
to  the  sum  of  the  partial  differences  of  latitude,  and  the  whole 
departure  is  equal  to  the  sum  of  the  partial  departures." 

The  word  sum  is  used  in  its  algebraic  sense,  that  is  to  say, 
if  1N  is  the  sum  of  the  northerly  differences  of  latitude  and  ls 


228 


NAVIGATION 


the  sum  of  the  southerly  differences  of  latitude,  then 
I  =  1N  ~  ls  and  is  of  the  same  name  as  the  greater  ;  and  if  pw 
is  the  sum  of  westerly  and  pE  of  easterly  departures, 
p  =  pw~pE  and  is  of  the  same  name  as  the  greater. 

The  traverse  table  referred  to  under  plane  sailing  greatly 
facilitates  the  computation. 

Having  found  the  resultant  I  and  p,  the  course  and  dis- 
tance made  good,  or  the  resultant  course  and  distance,  are 
gotten  from  the  formulae 


or  by  inspection  from  Table  2. 

The  traverse  may  run  irregularly,  and  into  higher  lati- 
tudes than  the   latitudes  of 

] 


— <--*-)• 


the  extremities  of  the  dis- 
tance made  good,  so  that  the 
departure  of  this  course  and 
distance  made  good  may  not 
be  the  same  as  the  sum  of  the 
partial  departures;  hence,  if 
necessary,  separate  the  trav- 
erse into  two  or  more  parts, 
and  calculate  for  each  part 
separately.  However,  if  the 
traverse  does  not  go  into  too 
high  latitudes,  the  error,  like- 
ly to  arise,  may  be  considered 
immaterial  in  an  ordinary 
day's  run. 

Graphic  explanation  of 

traverse  sailing.  —  To  further 

i  •       .1  •  i          j? 

explain     the     principles     of 

traverse  sailing,  let  W'E'gb  represent  a  portion  of  a  Mer- 
cator  chart;  let  A,  located  on  a  meridian  NS  and  a  parallel 


GRAPHIC  EXPLANATION  229 

of  latitude  WE',  be  a  place  sailed  from,  and  F  a  place  arrived 
at,  after  sailing  successively  from  A  to  B,  to  C  (directly  East), 
to  D  (directly  North),  to  E,  and  to  F  (Fig.  68).  Let  merid- 
ians and  parallels  be  drawn  through  each  point  of  the  traverse ; 
the  triangles  thus  formed  and  the  difference  of  latitude  and 
departure  corresponding  to  each  distance  are  indicated  in  the 
figure. 

.  _  C  Diff.  of  Lat.  =  Ac. 
For  distance  AB  \  _.  „       ,  _ 

|  Dep.  =  cB  =  lib. 

„ n  (  Diff.  of  Lat.  =  0. 
For  distance  BC  4~  ^n       , 

(  Dep.  —  BC  —  Im. 

nr.  f Diff.  of  Lat.  =  CD  =  cd. 
For  distance  CD  \  ~ 

(Dep.  =  0. 

-n      T  x         T->  T7  (  Diff-  °f  Lat.  =  Dn  =  de. 
For  distance  DE  1  ^ 

J  Dep.  =  nE  =  mg. 

•Ei         J'4.  T?T?   f  Diff«    °f  Lat'   =  E9    ~    eJl' 

For  distance  EF  \  _ 

{  Dep.  =  ^. 

The  course  made  good  is  hAF,  distance  made  good  AF,  the 
corresponding  difference  of  latitude  Ah,  and  departure  JiF. 

Eegarding  directions  towards  the  top  of  page  and  to  the 
right  hand  as  positive,  differences  of  latitude  towards  the  top 
of  page  (in  this  case  North)  are  +,  towards  the  bottom  (or 
South)  are  — ,  departures  to  the  right  (or  East)  are  +,  to 
the  left  (or  West)  are  — . 

It  will  be  seen,  by  examining  the  figure,  that  All  and  JiF 
are  respectively  the  algebraic  sum  of  all  the  differences  of  lati- 
tude and  departures  corresponding  to  the  several  courses  and 
distances  sailed. 

Proof : 
I  =  +  Ac  +  (cd  — de)  +  eh  =  +  Ac  +  ce  +  eh  =  +  Ah. 

p  =  —  Jib  +  bm  +  mg  —  gF  =  —  Jim  +  mF  =  +  JiF. 

Sources  of  data.— -The  navigator  will  find  in  the  ship's  log 
book  the  latitude  and  longitude  of  the  point  of  departure,  the 


230  NAVIGATION" 

compass  courses  and  distances  sailed,  and  correction  for  lee- 
way, if  any;  then  taking  from  the  chart  or  tables  the  varia- 
tion of  the  locality,  and  from  the  deviation  table  the  devia- 
tions for  the  various  compass  courses  steered,  he  will  have 
the  data  for  working  the  traverse. 

If  in  a  region  of  known  currents,  he  must  allow  for  the  set 
and  drift  as  explained. later  (see  Arts.  129-131). 

The  preparation  of  the  traverse  form  and  data. — (1)  In 
case  a  departure  course  enters  into  the  computation,  the  com- 
pass bearing  of  the  point  of  departure  is  corrected  for  varia- 
tion, and  for  the  deviation  due  to  ship's  head  when  the  bearing 
was  taken,  and  the  reversed  true  bearing  thus  obtained  is 
entered  in  the  column  of  true  courses,  the  distance  in  the 
column  of  distances,  thus  forming  the  first  course  and  distance 
of  the  tabulated  form. 

(2)  Each  compass  course  is  corrected  for  variation,  devia- 
tion, and  leeway,  and  the  result  entered  in  the  form  under  the 
head  of  true  course,  the  sum  total  of  distances  run  on  each 
true  course  being  placed  opposite  that  course  in  the  distance 
column. 

(3)  Enter  Table  2,  Bowditch,  find  each  true  course  (7N  at 
top  or  bottom  of  page,  and  for  each  true  course  and  distance 
find  the  corresponding  differences  of  latitude  and  departure, 
placing  them  in  their  respective  columns,  opposite  the  courses ; 
the  difference  of  latitude  being  placed  in  the  "N."  column 
when  the  course  is  northerly,  in  the  "  S."  column  when  south- 
erly; the  departure  being  placed  in  the  "E."  column  when 
the  course  is  easterly,  in  the  "W."  column  when  westerly. 
When  distances  are  in  miles  and  decimals,  multiply  by  10  or 
by  100,  take  out  for  the  new  whole  number  the  desired  quan- 
tities, and  divide  them  by  the  multiplier  just  used.    Thus,  for 
29.3  take  out  I  and  p  for  293  and  divide  by  10.     Where  the 
course  is  between  two  given  degrees,  first  find  Z  and  p  for  each 
and  interpolate. 


TRAVERSE  FORM  AND  DATA  231 

Add  up  the  "  diff .  of  lat."  and  "  departure  "  columns ;  take 
the  difference  between  the  northings  and  southings  to  which 
give  the  name  of  the  greater;  do  the  same  for  the  E.  and  W. 
departures;  these  resulting  differences  are,  respectively,  the 
difference  of  latitude  and  departure  of  the  resultant  course. 

(4)  Then  look  in  Table  2  and  turn  to  that  page  on  which 
can  be  found  coincidently  in  the  lat.  and  dep.  columns  the 
above  mentioned  resultant  difference  of  latitude  and  departure. 
The  angle  at  the  top  or  bottom  of  page,  as  the  case  may  be,  will 
be  the  course  made  by  D.  K.,  estimated  from  the  North  point 
to  the  right,  the  particular  quadrant  being  determined  by  a 
consideration  of  the  resulting  differences  of  latitude  and  de- 
parture.   The  course  (7N  will  be  taken  out  in  the  1st,  2d,  3d,  or 
4th  quadrant  according  as  the  co-ordinates  /  and  p  show  it  to 
be  in  the  general  direction  of  NE.,  SE.,  SW.,  or  NW.,  re- 
spectively. 

On  the  same  line  with  the  difference  of  latitude  and  de- 
parture, in  the  Distance  Column,  will  be  found  the  distance 
made  by  D.  E. 

(5)  When  the  exact  values  of  /  and  p  are  not  found  together 
on  any  page,  turn  to  that  page  where  they  are  found  to  agree 
most  closely,  and  interpolate  between  the  degrees  of  this  and 
an  adjoining  page  for  the  course,  and  also  between  the  dis- 
tances involved  till  close  approximations  to  the  real  values  of 
C  and  D  are  reached. 

(6)  The  compass  being  graduated  from  0°  at  North,  around 
to  the  right  through  360°,  attention  is  called  to  the  fact  that 
the  1st,  2d,  3d,  and  4th  quadrants  are  respectively  the  NE., 
SE.,  SW.,  and  NW.  quadrants ;  and,  for  the  proper  marking  of 
7  and  p,  that  the  course  (7N  is  northerly  in  the  1st  and  4th  quad- 
rants, southerly  in  the  2d  and  3d,  easterly  in  the  1st  and  2d, 
and  westerly  in  the  3d  and  4th. 


232 


NAVIGATION 


Ex.  25. — On  April  3,  1905,  at  1  p.  m.,  took  departure,  Cape 
Henry  Lighthouse  (Lat.  36°  55'  35"  N.,  Long.  76°  00'  27" 
W.)  bearing  (p.  s.  c.)  293°,  distant  10  miles,  ship's  head 
East  (p.  s.  c.)  ;  deviation  +  3°,  variation  from  chart  —  6°. 
Thence  ran  till  8  a.  m.  next  day  as  follows.  Required  the 
course  and  distance  by  D.  R.  from  the  lighthouse,  and  the 
latitude  in. 


Courses  (p.  s.  c.). 

Distance. 

Dev. 

Leeway. 

Wind. 

73° 

60 

+  3° 

3° 

Nly 

118 

20 

+  6 

3 

Nly  and  Ely 

160 

10 

+  3 

3 

Ely 

319 

10 

-6 

3 

Nly  and  Ely 

26 

28 

+  3 

3 

Sly  and  Ely 

To  illustrate  the  application  of  the  various  corrections,  each 
course  will  be  considered  separately,  and  corrections  applied 
one  at  a  time,  though  in  practice  the  algebraic  sum  is  applied 
mentally. 


Departure  Course. 

Bearing  of  Lt.  (p.  c.)  293° 
Deviation     +3 

Course  (p.  c.) 
Leeway 
Course  thro'  water 
Deviation 
Magnetic  Course 
Variation 
True  Course 

1st 
Course. 

2d 
Course. 

73° 
+    3 

118° 
+  3 

76 

+    3 

121 
-1-6 

Magnetic  Bearing  296 

Variation              —6 

79 
-   6 

127 
-6 

True  Bearing  of  Lt  290 
Departure  Course         ...  .110 

73 

121 

Course  (p  c  ) 

3d  Course. 

4th  Course. 

5th  Course. 

160° 
+  3 

319° 
-3 

26° 
-3 

Leeway  

Course  through  Water.  .  .  . 
Deviation  

163 

+  3 

316 
—6 

23 
+  3 

Magnetic  Course 

166 
—6 

310 
-6 

26 

—6 

Variation  

True  Course    

160 

304 

20 

FORM  OF  WORK 


233 


E       -3 


o    ft 


.2     .    * 


§  S| 

G  H    "S 

5  O 

2  .2 


I? 

3  <n 

O  Q, 
OC? 


<  -«*i  i—  i  •«*<    .0 


CO  C 


I?* 

.2  ^  " 


234 


NAVIGATION 


PARALLEL    SAILING. 

127.  Plane  sailing  has  dealt  only  with  courses  and  dis- 
tances sailed,  found  the  resulting  differences  of  latitude  and 
departure,  and  given  to  the  navigator  the  course  and  distance 
made  good  and  the  latitude  arrived  at,  without,  however, 
giving  him  his  longitude.  To  find  the  longitude,  a  relation 
must  be  established  between  it  and  the  departure.. 

Now  departure  is  measured  on  a  small  circle  of  the  sphere 
paralled  to  the  equator,  is  expressed  in  units  of  a  sea  mile,  and 
represents  the  distance  between  two  meridians  in  the  same 
latitude.  The  arc  of  the  equator,  intercepted  between  these 
meridians,  is  a  measure  of  the  angle  between  them  and  ex- 
presses the  difference  of  longitude. 

Before  the  days  of  accurate  chronometers,  ship  masters 
were  often  uncertain  as  to  their  longitude.  They  would  "  run 
down  their  latitude,"  that  is,  go  N.  or  S. 
till  they  reached  the  latitude  of  the  port 
to  which  they  were  bound,  then  steer  for 
the  port,  along  a  parallel  E.  or  W.  The 
distance  made  along  a  parallel  was  the  de- 
parture, and  it  was  then,  as  now,  the  func- 
tion of  parallel  sailing  to  connect  depart- 
ure with  its  corresponding  difference  of 
longitude.  Knowing  the  longitude  left, 
and  finding  by  an  application  of  the  prin- 
ciples of  parallel  sailing  the  difference  of 
longitude  made,  we  are  enabled  to  find 
the  longitude  arrived  at. 

In  Fig.  69,  EF  is  the  difference  of  longitude  between  me- 
ridians PE  and  PF  and  is  called  D.  BA  is  the  corresponding 
departure  in  the  latitude  of  B  and  A  and  is  called  p.  Now  D 


PARALLEL  SAILING  235 

and  p  are  similar  arcs  of  two  circles  and  therefore  are  propor- 
tional to  their  radii,  hence,  D  :  p  —  R  \  r,  and  p  =  -„-• 

Now  L  AOF  =  L  OAC  is  the  latitude  of  B  or  A,  and 
cos  L  =  -^ ;  therefore, 

p  =  D  cos  L;  or  D  =  p  sec  Z.  (119) 

Hence,  by  this  formula,  is  found  the  difference  of  longitude 
corresponding  to  a  given  departure  in  a  given  latitude. 

The  relation  of  parts  involved  in  parallel  sailing  are  shown 
in  the  triangle,  Fig.  70,  in  which  L  is  the  latitude,  p  the  de- 
parture, D  the  corresponding  difference  of  longitude,  the  de- 
parture being  in  sea  miles  and  longitude  in  minutes  of  arc. 

The  most  usual  cases  that  arise 
under  parallel  sailing  are  those  in 
which  the  departure  between  two 
places  in  the  same  latitude  is 
given  to  find  the  difference  of 
longitude,  or,  given  the  difference 


of  longitude  between  two  places  pIG>  70. 

in  the  same  latitude,  to  find  the 

departure;  though  sometimes  the  latitude  of  the  parallel  may 

be  required,  the  difference  of  longitude  and  corresponding 

departure  between  the  two  places  being  known. 

CASE  I. — Given  p,  to  find  D. 

Ex.  26.— K  ship  in  latitude  49°  35'  K  and  longitude  22° 
30'  W.  sails  due  South  (true)  65  miles,  then  due  East  (true) 
120  miles;  find  latitude  and  longitude  in. 

By  computation: 

L,  =  49°  35'  N  fZ2  =  48  30     sec  10.178741  ^  =  22°  30'  W 
I  =   1   05  SJ  p  =  120       log   2.07918  ID  =  3   01.1  E 

£1=48030/N[z>  =181.1     log   2.25792J^2=19028'.9W 

By  inspection:  Enter  table  2  with  latitude  as  a  course, 
find  p  -in  the  latitude  column,  and  opposite  in  the  distance  col- 


236  NAVIGATION 

unm  is  P.     It  may  be  necesary  to  interpolate  as  in  the  ex- 
ample.   The  Lat.  of  the  parallel  is  48°  30'  IsT. 

For  48°,  j9  =  119.8,  Z)  =  179     For  49°,  p  =  119.4,  Z)  =  182 
p  =  120.4,  .0  =  180  p-  120.1,  D=183 


jt?  =  120,     D=  179.33  p  =  120,      D  =  182.86 

Hence  for  Lat.  48°  30'  K,  and  p  =  120,  D  —  181.095. 
CASE  II.  —  Given  Df  to  find  p. 

Ex.  27.  —  A  ship  sails  on  a  parallel  of  latitude  41°  30'  S. 
from  A  in  longitude  18°  30'  E.  to  B  in  longitude  2°  10'  W. 
Find  the  distance  sailed  in  nautical  miles. 

Long,  of  A     18°  30'  E          L  =  41°  30'  ____  cos  9.87446. 
"         B       2    10  W          D  =  1240  .....  log  3.09342 


D  =  20°  40' W          p=  928'.7 log  2.96788 

=  1240' 

Ans.  928.7  miles. 

By  inspection :  Enter  table  2  with  Lat.  for  the  course,  find 
D  in  the  distance  column,  and,  opposite  in  the  latitude  column, 
will  be  found  p. 

When  D  is  greater  than  any  tabulated  distance,  pursue 
either  of  the  following  methods  which  are  given  in  full  to 
illustrate  the  use  of  the  traverse  tables. 

Divide  D  by  10,  find  corresponding  p  in  latitude  column, 
then  multiply  by  10,  thus : 

For  L  —  41°,  D  —  124;  p  =  93.6  1     By  interpolation  and 
L  =  42°,  D  =  124;  p  =  92.1  j  multiplication, 

For  L  =  41°  30',  D  =  1240;  p  =  928.5. 
A  closer  result  may  be  gotten  by  inspection  by  considering 
D  in  three  parts,  600  +  600  +  40. 

For  Lat.  41°,  D  =  600,  p  =  452.8 
D  —  600,  p  =  452.8 
D  —  40,  p  =  30.2 


D  =  1240,  p  =  935.8 


EXAMPLES  FOR  EXERCISE  237 

For  Lai  42°,  D=  600,  p  =  445.9 
D  =  600,  p  =  445.9 
P  =  40,  p  =  29.7 


P  =  1240,  p  =  921.5 
By  interpolation,  for  Lat.  41°  30'  S.,  D  =  1240;  p  =  928.65. 

CASE  III.—  To  find  the  latitude,  given  p  and  D. 

Ex.  28.—  A.ship  in  longitude  45°  10'  W.  sails  due  W.  (true) 
186.8  miles,  and  is  then  in  longitude  48°  58'  W.  Find  the 
latitude. 

By  computation  :  cos  L  •=. 


A2  =  48°  58'  W          p=  186.8  .......  log  2.27138 

A!  =  45    10  W          D  —  228  ........  log  2.35793 


D  —    3°48'W         L  =  34°59'N cos  9.91345 

=       228  W 

By  inspection:  Turn  to  that  page  of  table  2  on  which 
D  =  228  is  found  in  the  distance  column  and  p  =  186.8  in 
the  diff.  of  lat.  column,  opposite  D.  The  angle  at  the  top  or 
bottom  of  page,  as  the  case  may  be,  will  be  the  latitude.  If 
exact  coincidence  of  D  and  p  are  not  found  on  a  given  page, 
then  the  angle  must  be  found  by  interpolation. 
In  this  example  Lat.  is  found  to  be  35°  N". 

Examples  Under  Parallel  Sailing. 

Ex.  29.— A  ship  in  latitude  38°  N.  sailed  due  West  till  she 
changed  her  longitude  5°.  What  distance  did  she  sail? 

Ans.  236.4  miles. 

Ex.  80.— A  ship  in  Lat.  40°  K,  Long.  160°  W.,  sails  due 
East  till  her  longitude  is  150°  30'  W.  Find  by  inspection  the 
distance  sailed.  Ans.  436.6  miles. 


238  NAVIGATION 

Ex.  31. — How  far  must  a  ship  sail  due  East  in  Lat.  60°  N. 
to  change  her  longitude  5°  ?  Ans.  150  miles. 

Ex.  32.— From  a  place  in  Lat.  30°  N.,  Long.  50°  20'  W.,  a 
ship  sails  due  West  240  miles,  then  due  N".  240  miles,  and 
due  E.  240  miles.     Find  Lat.  and  Long,  in  by  inspection. 
Ans.  Lat.  34°  K;  Long.  50°  07'.6  W. 

Ex.  33. — A  ship  in  Lat.  60°  N.  sails  due  West  75  miles. 
How  much  does  she  change  longitude?  Ans.  2°  30'  W. 

Ex.  34.— A  ship  in  Lat.  38°  K,  Long.  159°  10'  W.  sails  due 
E.  405  miles.  Find  Lat.  and  Long.  in. 

Ans.  Lat.  38°  K;  Long.  150°  36'  W. 

Ex.  35. — Two  ships  in  Lat.  35°  N".,  distant  from  each  other 
150.7  miles,  sail  due  North  at  the  same  speed  for  300  miles. 
Find  by  inspection  how  much  closer  they  are  at  the  end  of 
run.  Ans.  9.7  miles. 

Ex.  36. — Find  by  inspection  in  what  latitude  the  length  of 
a  degree  of  longitude  will  be  46  miles.  Ans.  40°. 

Ex.  37. — Two  ships  are  steaming  due  East  at  the  same 
speed.  B  changes  longitude  twice  as  fast  as  A,  who  is  in  the 
20th  parallel  of  1ST.  latitude  and  to  southward  of  B.  Find  B's 
latitude  by  computation.  Ans.  61°  58'  31"  K 

MIDDLE  LATITUDE   SAILING. 

128.  In  plane  sailing,  the  assumption  was  made  that  the 
earth  was  an  extended  plane,  and,  though  this  assumption  was 
false,  the  errors  for  small  distances  were  considered  imma- 
terial. Were  the  earth  an  extended  plane,  the  departure 
would  be  the  same  in  both  latitudes  left  and  arrived  at.  It 
has  been  shown  that  the  departure  is  approximately  that  of 
the  middle  latitude. 

In  parallel  sailing,  the  earth  was  regarded  as  a  sphere  and  a 
relation  was  established  between  departure  and  difference  of 
longitude. 


MIDDLE  LATITUDE  SAILING 


239 


Now,  combining  the  principles'  of  plane  and  parallel  sail- 
ing, we  have  middle  latitude  sailing,  which  finds  the  difference 
of  longitude  corresponding  to  a  departure 
measured  in  the  middle  latitude,  and,  by 
partially  nullifying  the  false  assumptions 
of  plane  sailing,  gives  a  nearer  approxi- 
mation to  true  results.  A  still  nearer  ap- 
proximation to  the  truth  may  be  gotten 
by  applying  from  Bowditch  a  correction 
to  the  middle  latitude,  and  considering 
the  departure  measured  on  this  corrected 
parallel.  However,  if  necessary  to  do  this, 
it  would  be  better,  after  finding  C  and  L2 
by  plane  sailing,  to  find  D  by  Mercator 
sailing,  explained  later  on. 

The  relations  of  the  quantities  involved  in  middle  latitude 
sailing  are  shown  in  Fig.  71  by  combining  the  triangles  of 
plane  and  parallel  sailings,  regarding  the  departure  as  meas- 
ured in  the  latitude  of  the  middle  parallel. 


FIG.  71. 


Let  Ln  = 


the  middle  latitude,  then 


I  =  d  cos  C,  p  =  d  sin  0. 
D  =  p  sec  L0  —  d  sin  C  sec  L0  —  I  tan  C  sec  Z/0. 


tanC  = 


D  cos  Ln 


(120) 


Lo=  A±_A, 

L2  =  A  +  I  A2  =  A,  +  D. 

When  not  advisable  to  use  M.  L.  sailing. — The  results  got- 
ten by  using  middle  latitude  sailing  are  more  accurate  in  low 
latitudes,  less  so  in  high  latitudes,  and  the  inaccuracy  is 
greater  the  greater  the  difference  of  latitude,  or  for  a  given 
distance  sailed,  the  smaller  the  course.  Hence  when  the  lati- 
tudes are  high  (over  50°  N.  or  S.),  or  course  small  with  a 
large  distance  producing  large  differences  of  latitude,  it  is 


240  NAVIGATION 

better  not  to  use  middle  latitude  sailing,  but  to  use  Mercator 
sailing. 

It  is  not  advisable  to  use  middle  latitude  sailing  when  the 
places  left  and  arrived  at  are  on  different  sides  of  the  equator, 
unless  the  two  parts  of  the  track  on  opposite  sides  of  the 
equator  are  treated  separately,  except  in  the  case  where  the 
distance  each  side  is  so  small  that  the  departure  is  practically 
equal  to  the  difference  of  longitude. 

In  ordinary  practice  examples  under  middle  latitude  sail- 
ing come  under  one  of  the  two  following  cases.  For  other 
variations,  however,  it  is  only  necessary  to  draw  a  figure  show- 
ing the  relation  of  the  parts,  and  to  use  that  formula  which 
will  give  the  unknown  from  the  known  parts. 

Case  I. — Given  the  course  and  distance  sailed  from  a  place 
of  known  latitude  and  longitude,  to  find  the  latitude  and 
longitude  arrived  at. 

Ex.  38.— A  ship  in  Lat.  36°  40'  S.,  Long.  48°  40'  W.  sailed 
38°  (true)  150  miles.  Find  Lat.  and  Long.  in.  Solution  by 
computation : 

o      /     // 

d—  150 log  2.17609 log'2.17609       Ll  —  36  40         S 

tf— N38°E..          ..cos  9.89653.  ...  sin  9.78934  Z  r=    1   58  12  N 


Z=118'.2N log  2.07262  Z2  —  34  41  48   S 

p      log  1.96543      LQ  =  35  40  54    S 

L0  =  35°  40'  54"  S secO.09030       ^  =  48  40         W 


—  113'.  69  E  .....................  log  2.05573       D  —    1  53  41  E 


By  inspection:  Enter  Table  2  with  course  38°,  opposite  150 
in  distance  column,  find  /=  118.2  in  Lat.  column  and  p=92.3 
in  Dep.  column.  Then  with  the  middle  latitude  35f  °  as  a 


MIDDLE  LATITUDE  SAILING  241 

course  find,  by  interpolation,  opposite  p  in  Lat.  column,  the  D 
in  distance  column. 

For  Lat.  35°,  ^  =  92.3,  »  =  112.7  I   .  for  Lat   ^y     ^  =  92.3,  D  =  113.7 
"        36°,  p  —  92.3,  D  z=  114.1  | 

o   /   //  o   /  // 

L!  =  36  40    S         A!  =  48  40  00  W 
Zrz  1  58  12  N         D—  1  53  42  E 


£2  =  34  41  48  S         A2  =  46  46  18  W 
A,  =  35%°    S 

Case  II. — To  find  the  course  and  distance  between  two  posi- 
tions of  known  latitude  and  longitude. 

Ex.  39. — Find  the  course  and  distance  from  Lat.  43°  03' 
24"  K,  Long.  5°  56'  30"  E.  to  Lat.  39°  26'  42"  N.,  Long. 
0°  23'  00"  W. 

By  computation: 

O       I      It  O      I      ff  O       I      It 

LI=  43  03  24  N  AI  =  5  56  30  E  LI  =  43  03  24  N 

I/2  =  39  26  42  N  A2  =  0  23  00  W  I/2  =  39  26  42 


82  30  06 
I  =  216'.7  =   3  36  42  S  D  =  379'.5  =  6  19  30  W  L0  =  41  15  03  N 

D  =  379.5 log  2.57921 

I/o  =  41°  15'  03" cos  9.87612 

p log  2.45533  . .  log  2.45633 

I  =  216.7 log  2.33586 log   2.33586 

0=  S  52°  47'  W  (tfN=232°  47') tan  0.11947 sec  10.21837 

d  =  358.28  miles log  2.55423 

By  inspection.— Enter  Table  2  with  L0  =  41J°  as  C,  find 
corresponding  to  D  379.5  in  distance  column,  p  =  285.3  in 
Lat.  column,  thus 

For  Lat.  41°,  D  —  379.5;     p  —  286.4  |  Therefore 
For  Lat.  42,     D  — 379.5;     ,p:=282.    )Z0  =  41%'0,  D  =  379.5,  ^  =  285.3 

Then  find  corresponding  to   I  —  216.7   and  p  =  285.3   the 
course  and  distance  thus: 

For  p  —  285.3,        I  —  222.9;       d—  362.          CN  =  232°  |  Therefore  by 
p—  285.3,        Z  =  215.;         d—  357.2        CN  =  233°j"      interpolation 

For  ^  =  285.3,        I  —  216.7;        rf  =  358.4        <7N  —  232%° 


242 


NAVIGATION 


Again  attention  is  called  to  the  fact  that/if  the  difference  in 
latitude  is  large,  the  assumption  that  the  departure  is  properly 
measured  in  the  middle  latitude  is  not  strictly  correct  ;  and,  if 
greater  accuracy  is  desired,  a  correction  from  Bowditch  must 
be  applied  to  the  middle  latitude  to  obtain  the  proper  parallel 
on  which  to  take  the  departure  (see  Art.  133).  However, 
it  is  just  as  easy  and  more  correct  to  use  Mercator  sailing. 

When  the  two  places  considered  are  on  opposite  sides  of  the 
equator,  no  sensible  error  will  be  made  in  the  case  of  an  ordi- 
nary day's  run,  which  will  seldom  exceed  400  miles,  by  taking 
the  difference  of  longitude  equal  to  the  departure.  If  the 
distance  is  great,  use  Mercator  sail- 
ing, except  when  the  course  is  large 
(more  nearly  East  or  West),  in 
which  case  use  middle  latitude  sail- 
ing (see  Art.  132). 

However,  when  the  distance  be- 
tween two  places,  one  in  North  lati- 
tude and  the  other  in  South  latitude, 
is  great,  and  it  is  desired  to  use 
middle  latitude  sailing  in  finding 
the  difference  of  longitude,  the  two  portions  of  the  track 
on  different  sides  of  the  equator  may  be  treated  sepa- 
rately. Thus  in  Fig.  72,  let  the  coordinates  of  the  place  A 
be  L!  ,  A!  ,  and  those  of  the  place  C  in  the  opposite  hemi- 
sphere be.  L2  ,  X2. 

The  track  A  C  is  divided  by  the  equator  EQ  into  two  parts, 
AB  and  EC. 
For  AB  we  have 


FIG.  72. 


i  —  L:  tan  (7,  and  neglecting  ALX  , 
=  D,  =  Pl  sec  ^  =  L,  tan  0  sec^1 


EXAMPLES  FOR  EXERCISE 
For  BC  we  have 


243 


p2  =  ( — )  L2  tan  C,  and  neglecting  &L2 , 
BE  =  D2  =  p2  sec  -£  =  ( — )  L2  tan  C  sec  -^, 

whence  QE  or  D  =  D±  +  D2 . 

Therefore,  for  this  case  we  have  the  following  formulae: 
I  =  d  cos  C, 


=  LI  tan  C  sec  -^ 


(121) 


D2  =  ( — )  L2  tan  C  sec  ± 

\  \  I      T\ 

Ao  —  A!  ~pX/« 

Instead  of  the  middle  latitude  £  L±  and  ^  L2 ,  we  may  for 
greater  precision  use  (£  L±  +  A  .L±)  and  (J  L2  -f-  A  L2). 

Examples  in  Middle  Latitude  Sailing. 
(By  inspection.) 

Ex.  40.— From  7^  49°  28'  30"  K,  At  0°  03'  15"  B.,  sailed 
312°  (p.  s.  c.)  36  miles,  variation  —20°,  deviation  —2°. 
Find  by  D.  E.  L2  and  A2. 

Ans.  £2  =  49°  40'  48"  N. 
A2  =    0     48    51    W. 

Ex.  41.— From  £±  48°  20'  29"  K,  A±  5°  07'  48"  W.,  sailed 
257°   (p.  s.  c.),  22.2  miles,  variation  — 20%  deviation  — 3°, 
thence  232°  (p.  s.  c.),  216.5  miles,  variation  —  20°,  deviation 
-1°.    Find  L2  and  A2. 

Ans.  L2  =  45°  01'  55"  K 
A2=:    8     16    30    W. 

Ex.  42.— A  ship  leaving  Lat.  49°  50'  N.,  Long.  10°  16'  W., 
sails  to  the  southward  and  westward  till  her  departure  is  188 


244  NAVIGATION 

miles  and  the  latitude  reached  is  47°  28'  N".    Find  the  course, 
distance,  and  longitude  in. 

fCourse  CN  =  233°. 
AnsA  Distance  236. 

[\2  =  15°  00'  42"  W. 

Ex.  43.— A  ship  sails  from  L±  24°  23'  S.,  \^  100°  30'  E., 
and,  by  observations  the  next  day,  finds  her  position  to  be 
25°  43'  12"  S.,  104°  52'  38"  E.  What  was  her  true  course 
and  distance?  Ans.  <7N  =  108.°6. 

Distance  251.9  miles. 

Ex.  44- — Find  by  computation  the  true  C  and  d  from  Ll 
23°  00'  N".,  A!  109°  55'  W.,  to  L2  35°  30'  K,  \,  139°  45'  E.. 
(without  correcting  middle  latitude). 

Ans.  0N  =  277°  25'  12". 

Distance  5807.5  miles. 

CUBBENT  SAILING. 

129.  A  current  may  be  defined  as  a  body  of  water  moving 
steadily  in  one  direction. 

The  set  of  a  current  is  its  course,  or  the  direction  in  which 
it  is  moving. 

The  drift  is  the  distance  the  current  sets  a  ship  in  the  time 
considered.  Thus  the  drift  in  20  hours  being  10  miles,  the 
drift  per  hour,  -J  mile,  would  be  more  properly  called  the  rate. 
When  the  rate  per  hour  is  known,  the  drift  for  any  given  time 
is  easily  found. 

When  a  ship  sails  directly  with  or  directly  against  a  current, 
her  motion  is  increased  or  retarded  by  the  amount  of  the  drift 
in  the  interval. 

When  a  ship  sails  obliquely  to  a  current,  her  motion  may 
be  accelerated,  or  retarded,  according  to  the  angle  between 
the  course  of  the  ship  and  the  set  of  current ;  and  the  distance 
made  good  is  the  diagonal  of  a  parallelogram  of  which  one 
side  is  the  distance  made  in  the  direction  of  the  keel  and  the 


CURRENT  SAILING 


245 


other  side  the  distance  the  ship  is  carried  by  the  current  in 
the  direction  of  the  set,  in  the  same  interval  of  time.  This 
resultant  direction  is  in  accordance  with  Newton's  first  and 
second  laws  of  motion. 

Current  sailing. — Current  sailing  may,  therefore,  be  defined 
as  the  means  of  finding  the  course  and  distance  made  good 
when  a  ship's  motion  is  affected  by  tides  or  currents,  or  a 
course  to  be  steered  to  make  good  a  given  course. 

Problems  in  current  sailing. — There  are  two  general  cases 
in  practice. 

Case  I. — Given  a  course  steered  and  distance  run,  to  find 
course  and  distance  made  good  through  a  current  of  known 
set  and  rate. 

Ex.  45. — In  Fig.  73,  let  MM'  be  the  meridian  of  a  place  A 
in  North  latitude  and  let  it  be  assumed  that  a  ship,  leaving  A, 
steers    210°     (true)     8    knots    per 
hour,    through     a    current    setting 
her  to  the  eastward   (true)  2  miles 
per  hour.     Lay  off  AB  =  8   miles 
in   the    direction    210*,   the    speed 
per  hour  of  the  ship  on  her  course. 
Lay  off  AC  =  2  miles  in  the  direc- 
tion East,  the  drift  and  set  of  cur- 
rent in  the  same  interval  of  time. 
Complete     the     parallelogram     by 
drawing  BD  and  CD,  and  join  AD. 
By  the  principle  of  "the  composi- 
tion of  forces,"  the  ship  at  the  ex- 
piration of  one  hour  will  be  at  D}  having  been  moved  along 
the  diagonal  AD  under  the  joint  action  of  two  forces,  her 
own  propelling  force  and  that  of  the  current.     The  result 
under  the  joint  forces  is  the  same  as  if  each  force  had  acted 
in  succession,  that  is,  as  if  the  ship  had  gone  from  A  to  B 


FIG.  73. 


246  NAVIGATION 

tinder  her  own  propelling  force  uninfluenced  by  current,  had 
then  stopped,  and  been  swept  from  B  to  D  by  the  current. 

In  the  particular  diagram  (Fig.  73)  AD  is  the  distance 
made  good;  MAD,  the  course  made  good  (from  IS]",  to  right). 

Solution  by  construction. — Having  a  Mercator  chart,  it  is 
only  necessary  to  lay  off  AB  and  AC  from  the  known  position 
A  in  the  proper  directions,  complete  the  parallelogram  as  above 
explained,  then  measure  the  angle  MAD  and  the  distance  AD. 

Solution  by  trigonometry. — To  solve  by  trigonometry,  make 
a  rough  sketch  to  show  the  conditions.  Referring  to  Fig.  73, 
we  have  the  angle  ABD  =  60°,  AB  —  8,  BD  =  2,  and  it  is 
required  to  find  L  BAD  and  AD.  Then  the  course  made  good 
OT/.MAD  =  210°  —  BAD  =  210°  —A.  Since  £ABD  =  60, 
A+D  .  180°  —  60°  120°  _  fino 

—      ——       :__ 


a  =  BD, 

From  plane  trigonometry  we  have,  as  ^  &  =  AD, 

d  =  AB, 

D  —  A      d  —  a      ,  B       6       ,  oAO 

tan  — H —  =  T— —  cot  -x  =   --  cot  30°. 


6 log    0.77815 

80° log  cot  10.23856 

10 ar.  co      9.00000 


-^46°  06'  07"  tan  0.01671 

*  ,4  ~    13  5o  Do 

To  find  AD  =  b,        b~d-  ^^=  8  sin  60°  cosec  106°  06'  07" 

o        t       it 

8    log  0.90309  210 

60° log  sin          9.93753  A—    135353 

106°  06'  07"  log  cosec    10.01738         MAD—  196  06  07 
6  —  7.211 log  0.85800 

The  course  made  good  is  CN  =  196°  06'  07".  Distance  made 
good  per  hour  =  7.211  miles. 


CURRENT  SAILING 


'247 


Solution  by  the  traverse  table. — It  is  a  simpler  plan,  how- 
ever, to  consider  the  course  sailed  and  the  set  of  the  current 
as  two  separate  courses  in  a  traverse  as  below.  Though  ap- 
proximate, results  will  be  sufficiently  correct. 


Diff.  lat. 

Departure. 

True 
Courses. 

Dist. 

N 

S 

E 

W 

210° 

8 

6.9 

4.0 

90° 

2 

.... 

.    .  . 

2.0 

6.9 

2.0 

4.0 

2.0 

2J) 

With  Z  =  6.9  )  Course  made  good  C7N=zl960 


Zr=6.9) 
p  =  2.0  J 


Distance    made   good  =  7. 2  miles. 


In  the  example  worked  above,  the  course  and  distance  of 
the  ship,  and  set  and  drift  of  the  current  were  given  for  one 
hour  only,  but  the  principle  holds  good  for  any  example  in 
which  the  set  and  drift  of  current  for  a  given  time  may  be 
taken  as  a  course  and  distance  in  a  traverse. 


248 


NAVIGATION 


Ex.  46.— A  ship  in  Lat.  40°  30'  N.,  Long.  48°  05'  W.,  at 
noon  on  Jan.  10,  sailed  till  noon  Jan.  11,  240°  (p.  s.  c.) 
223  miles.  Var.  —  24°,  dev.  —  2°.  A  current  sets  95°  (true) 
0.75  of  a  mile  per  hour.  Find  (7N  and  d  made  good,  Lat. 
and  Long.  in. 


Course 
(p.c.) 

Var. 

Dev. 

True 
Course 

Dist. 

3 

E 

W 

240° 

—24° 

—2° 

214° 

223 

184.9 

124.7 

Current 

95 

18 

1.6 

17.9 



0        /         ff 

L^  =  40  30  00  N 
I  =    3  06  30  S 

0         /          tf 

\  =  48  05  00  W 
D  -    2  17  30  W 

186.5 
Z  =  186.5 

17.9 

124.7 
17.9 

^=fcl06.8 

L3  =  37  23  30  N 
L0  =  38  56  45  N 

\  =  50  2^  30  W                 D  =  137.5  W 

~     .  ,.       (  Course  made  good  CfN=209°. 8. 

By  inspection  j  Digtance  madj  good  2U  9  mile8> 

Having  found  I  and  p,  the  course  and  distance  might  be 
gotten  from  the  formulas 

tan  C  = 


d  =  I  sec  G. 
However,  the  result  by  inspection  is  sufficiently  close. 

Case  II. — Given  the  ship's  speed  per  hour  and  the  bearing 
of  a  port  or  destination,  find  the  course  to  be  steered,  through 
a  current  of  known  set  and  rate,  in  order  to  keep  that  port, 
or  point  of  destination,  on  the  same  bearing. 

Solution  by  construction. — Let  N8  (Fig.  74)  be  a  meridian 
passing  through  the  place  A,  the  point  of  departure  on  the 
chart.  Let  AB  be  the  bearing  of  the  port,  or  the  direction 
to  be  made  good.  Draw  AB  on  the  chart.  Draw  AC  to  rep- 
resent the  set  and  hourly  rate  of  the  current.  With  extremity 
C  of  the  current  line  as  a  center,  and  with  a  radius  equal  to 


CURRENT  SAILING 


249 


the  ship's  speed  per  hour,  taken  in  the  same  units  as  the  rate 
of  current,  describe  an  arc  cutting  AB  in  D.  Then  AF,  drawn 
parallel  to  CD,  will  be  the  direction  the  ship  should  be  steered 
to  keep  the  port  on  the  same  bearing  AB. 

Ex.  47. — What  will  be  the  magnetic  course  from  a  point  off 
Key  West  to  make  Morro  Light  House  at  Havana,  the  mag- 
netic bearing  of  which  from  chart  is  197°  ?  The  passage 
is  across  the  Gulf  Stream,  setting  75°  (mag.)  2  miles  per 
hour.  Speed  of  ship  12  knots. 

By  construction.  —  Pro- 
ceed as  just  explained.  In 
Fig.  75,  AB  is  the  magnetic 
bearing  of  the  port  197°. 
Lay  off  AC  =  2  miles  in  the 
direction  75°. 


FIG.  74. 


PIG.  75. 


to  represent  the  rate  and  set  of  current.  With  C  as  a  center- 
and  radius  of  12  miles,  strike  an  .arc  cutting  AB  in  D.  Draw 
AF  parallel  to  CD.  NAF  is  the  course  to  steer. 

By  trigonometry. — By  a  rough  sketch  show  the  actual  state 
of  affairs,  as  in  Fig.  75.    From  G,  the  extremity  of  the  cur- 


250 


NAVIGATION 


rent  line,  drop  a  perpendicular  CK  on  the  bearing  line,  then 
—  ^  sin(75°  —  17°)=  2  sin  58°. 


From  the  traverse  table,  p  =  d  sin  C,  and  as 
KG  =  2  sin  58°,  therefore, 
1.7  =  2  sin  58°  =  12  sin  x°. 
1.7  =  12  sin  x°. 


Course  to  steer  =  L  SAF  +  180°  =  17°  +  8°  +  180° 
=  205°.  Magnetic  course  to  steer,  205°.  Or,  since  L  NAC 
=  75°  and  L  SAD  =  17°,  /  DAC  =  122°,  and  we  have, 

2:12  =  sin  x:  sin  122°. 
Hence  by  logs.,  x  =  8°  08'  ;  therefore,  course  =  205°  08'. 

Solution  by  the  traverse  table  and  traverse  sailing.  — 
Reverse  the  direction  of  the  current  and  consider  the  ship  to 
sail  from  C  to  A,  and  then  from  A  to  D,  finding  the  course 
from  C  to  D. 

Traverse  Table. 


Current    ( 
Reversed  1 


Courses 
(Mag.). 

Dist. 

Diflf.  of  Lat. 

Departure. 

N 

S 

E 

W 

197° 
255 

12 
2 

11.5 
0.5 

3.5 
1.9 

1=12                          p  =  5A 
Course  Cy  =  205°  (Magnetic) 

This  solution  by  traverse  sailing  is  sufficiently  correct  for 
all  practical  purposes,  but  is  theoretically  in  error,  as  the  dis- 
tance sailed  in  the  direction  of  the  port  AB  is  not  12  miles, 
but  less  than  12  miles,  which  is  the  distance  sailed  per  hour  in 
the  direction  of  the  course. 

Ex.  48. — Steaming  at  the  rate  of  8.5  knots  per  hour,  one 
wishes  to  make  good  a  course  76°  magnetic,  through  a  cur- 


CURRENT  SAILING  251 

rent  setting  12°  magnetic  3.5  miles  per  hour.    What  must  be 
the  magnetic  course?  Ans.  CN  =  97°  43'  16". 

Ex.  49. — A  port  bears  from  a  ship  359°    (mag.)   distant 

127  miles.    Steaming  at  16  knots  per  hour,  through  a  current 

that  sets  320°   (mag.)   at  the  rate  of  3  miles  per  hour,  find 

the  compass  course  to  make  the  port,  deviation  —  2°,  var. 

-  7°.    Find  also  the  time  occupied  in  making  the  voyage. 

Ans.  Compass  course  <7N  =  7°  46'  34". 
Distance  good  per  hour,  18.22  miles.    Time,  6h.97. 

130.  Current  from  noon  positions. — To  current  is  usually 
attributed  the  discrepancy  between  the  noon  positions  at  sea 
by  observation  and  by  dead  reckoning,  or,  at  any  instant,  the 
difference  between  the  position  by  dead  reckoning  and  one 
obtained  by  bearings  of  known  landmarks. 

The  distance  between  the  two  positions  divided  by  the  num- 
ber of  hours  elapsed  since  leaving  a  position,  assumed  to  be 
correct,  will  give  the  hourly  rate  of  the  current;  the  bearing 
of  the  position  by  observation  from  that  by  dead  reckoning 
being  the  set,  or  direction  of  the  current. 

It  must  not  be  forgotten,  however,  that  the  current,  thus 
computed  and  so  called,  may  be  due  to  careless  steering,  im- 
proper logging  or  determination  of  the  speed,  or  to  errors  of 
observation,  rather  than  to  any  real  motion  of  the  waters  of 
the  sea. 

Ex.  50. — On  April  10,  a  vessel's  noon  position  by  observa- 
tion was  Lat.  40°  44'  K,  Long.  47°  12'  30"  W.;  by  D.  K., 
Lat.  40°  37'  K,  Long.  46°  51'  48"  W.  Find  set  and  drift  of 
current  since  preceding  noon. 

Lat.  by  obs.  40°  44'  00"  N"       Long,  by  obs.  47°  12'  30"  W 
"         D.  E.  40    37  00   N  "        D.  R.  46    51  48   W 


1=  r        N  D  =  20'.7:=20'42"W 

0  =  40$ °  N  p  =  15'.7  W 

(Set,  294°. 

|  Drift,  17.2  miles  in  24  hours. 


252  NAVIGATION 

In  the, above  example,  since  the  position  by  observation  is 
to  the  northward  and  westward  of  that  by  dead  reckoning, 
or  account,  it  is  evident  that  the  ship  was  set  to  the  northward 
and  westward  by  the  current,  therefore  mark  I  as  1ST.,  and  D 
and  p  as  W.  For  the  middle  latitude  40f  °,  considered  as  a 
course,  find  D  in  the  distance  column  of  Table  2,  and  opposite 
D,  take  p  out  of  the  latitude  column.  With  I  and  p  find  the 
corresponding  C  and  d,  or,  in  other  words,  the  set  and  drift  of 
the  current. 

Ex.  51. — At  noon  on  Jan.  10,  the  ship's  position  by  observa- 
tion was  Lat.  25°  43'  12"  S.,  Long.  104°  52'  38"  E.  The  posi- 
tion by  D.  E.  from  previous  noon  was  Lat.  25°  52'  48"  S., 
Long.  104°  30'  24"  E.  Find  the  set  and  drift  of  current. 

Lat.  by  obs.  25°  43' 12"  S        Long,  by  obs.  104°  52' 38"  E 
"        D.E.  25    52  48    S  "         D.  E.  104    30  24  E 


I  —  9'.6  =  9'  36"  N  D  —  22'.23  =  22'  14"  E 

L0  =  25J°  S  p  —  20'.02  E 

f  Set,  64°.4. 
'   j  Drift,  22.2  miles  in  24  hours. 

In  this  example  the  true  position  is  to  the  northward  and 
eastward  of  that  by  account,  therefore  the  ship  was  set  to 
northward  and  eastward,  and  we  must  mark  I,  N. ;  D  and  p,  E. 

131.  Tidal  currents. — The  navigator  should  pay  careful 
attention  to  the  subject  of  tidal  currents,  and  shape  his  course, 
or  work  his  reckoning,  to  make  due  allowance  for  the  pos- 
sible set  and  drift,  in  all  localities  where  such  currents  have 
been  investigated.  Much  information  may  be  found  on  charts 
and  in  sailing  directions. 

Finding  from  the  tide  tables  the  times  of  high  and  low 
waters  at  places  along  a  coast,  it  may  often  be  possible  to 
make  allowance,  during  a  run  at"  such  times,  for  a  set  towards 
or  from  that  coast 


MERCATOR  SAILING  253 

When  the  wind  has  been  strong  and  steady  from  tme  direc- 
tion for  any  length  of  time,  a  current  may  be  produced  setting 
directly  to  leeward,  or  if  already  existing,  its  rate  may  be 
greatly  increased.  The  navigator  should  anticipate  and  en- 
deavor to  allow  for  its  effect. 

MERCATOR   SAILING. 

132.  It  has  been  shown  that  the  methods  of  middle  latitude 
sailing  are  sufficiently  exact  for  short  distances,  a  day's  run 
for  instance,  but  for  finding  the  difference  of  longitude  be- 
tween two  places  widely  separated  in  latitude,  or  for  finding 
the  course  between  two  such  places,  it  is  liable  to  great  error. 
To  avoid  such  errors  resort  is  had  to  Mercator's  sailing,  which 
is  based  on  principles  fully  explained  in  Art.  26,  and  applied 
in  the  construction  of  the  Mercator  chart,  and  which  furnish 
the  formula  that  gives  practically  correct  results. 

On  the  Mercator  chart,  the  meridians  are  drawn  parallel  to 
each  other  and  perpendicular  to  the  equator  and  parallels  of 
latitude,  so  arcs  on  parallels  are  represented  as  equal  to  the 
corresponding  arcs  of  the  equator,  or  differences  of  longitude ; 
in  other  words,  expanded  in  a  certain  ratio.  In  order  that 
the  rhumb  line  on  the  chart  may  make  the  same  angle  with 
each  meridian,  each  infinitesimal  element  of  latitude  must  be 
expanded  in  the  same  ratio  in  which  each  infinitesimal  ele- 
ment of  the  parallel  has  been  expanded.  If  the  earth  were  a 
perfect  sphere,  this  ratio  would  be  as  the  secant  of  the  latitude, 
but  as  the  earth  is  a  spheroid,  its  eccentricity  must  be  con- 
sidered. 

The  formula  from  Art.  27,  D  —  M  tan  C,  or 

D  =  tan  tf  [7915'.704  (loglo  tan  ( «  +  f .)-  ,  loglo  tang+|))] 

gives  the  relation  existing  in  Mercator  sailing  between  the 
constant  course  C,  the  latitude  L  of  a  point  on  the  loxodrome, 


254 


NAVIGATION 


and  the  difference  of  longitude  of  that  point  and  the  longitude 
in  which  the  loxo'drome  crosses  the  equator.  M  in  the  equa- 
tion is  the  augmented  latitude,  or  the  length  of  the  line  on 
the  Mercator  chart  indicating  the  latitude,  expressed  in  nau- 
tical miles,  according  to  the  scale  of  the  chart. 

If  it  is  desired  to  find  the  difference  of  longitude  between 
two  places  in  two  different  latitudes  Lt  and  L2 ,  substitute  in 
the  equation  successively  the  values  L±  and  L2 , 

letting  M x  be  the  augmented  latitude  corresponding  to  //± ; 
M 2  be  the  augmented  latitude  corresponding  to  L2 ; 
"        DI  be  the  difference  of  longitude  from  A  (Fig.  76), 
where  track  crosses  the  equator,  to  the  first 
point  in  latitude  L^ ; 
"        D2  be  the  same  to  second  point  L2 . 


LI  i 

E 


1 


FIG.  76. 


FIG.  77. 


In  Fig.  76,  let  EE'  be  the  equator,  L±  the  parallel  of  1st 
latitude,  L2  the  parallel  of  3d  latitude,  C  the  constant  course, 
then,  Z>±  =  M ±  tan  C. 
D2  =  M2  tan  C. 
D  =  D2  —  Di=  (M2  —  MJ  tan  C  =  m  tan  C.  (122) 

m  equals  the  meridional  difference,  or  augmented  difference 
of  latitude  between,  LJ.  and  L2 ,  and  is  the  length  of  the  line 
on  the  Mercator  chart  which  represents  the  true  difference  of 
latitude  between-  L±  and  L2 ,  expressed  in  nautical  miles,  ac- 
cording to  the  scale  of  the  chart. 


MERCATOR  SAILING 


255 


Table  3  of  Bowditch  is  a  table  of  meridional  parts  at  inter- 
vals of  one  minute  of  arc  up  to  80°,  compression  having  been 

taken  as  .     In  case  L±  and  L2  are  of  different  names, 

as  in  Fig.  77,  where  EE'  equals  the  equator,  lf±  and  M2  are  of 
different  names,  and  the  algebraic  difference  M2  —  M^  be- 
comes M  2  +  -^i-  Therefore, 

D=D2  +  D1  =  (M2  +  MJ  tan  C. 
Graphic  illustration  of  the  theory  of  Mercator  sailing. — 
Let  C'E'  represent  an  arc  of  the  equator,  and  CA  represent  a 
distance  sailed  on  a  rhumb  line  from  C  in  Lat.  _L±  to  A  in 
Lat.  L2 ,  shown  on  the  spheroid  in  Fig.  78,  and  on  the  Mer- 
cator chart  of  the  corresponding  limits  in  Fig.  79. 


Conceive  this  distance  to  be  subdivided  into  a  large  number 
of  small  parts,  and  the  elementary  triangles  to  be  formed  of 
which  the  corresponding  differences  of  latitude  ^ ,  12 ,  etc., 
are  represented  in  Fig.  78  by  Cn,  oi,  etc.,  and  the  departures 
by  no,  ir,  etc. 

Each  partial  departure  of  Fig.  78  is  represented  in  Fig.  79, 
a  section  of  a  Mercator  chart,  as  an  expanded  arc  equal  to  the 
corresponding  arc  of  the  equator,  no  equal  to  C'G,  tr  to  GK, 
etc. ;  so  that  the  departure  on  the  spheroid,  being  equal  to  the 


256  NAVIGATION 

sum  of  the  partial  departures,  is  expanded  into  the  corres- 
ponding difference  of  longitude  on  the  Mercator  chart. 

But,  in  order  that  the  angle  C  on  the  chart  shall  remain 
constant  and  equal  to  that  on  the  spheroid,  and,  that  the  simi- 
larity of  the  corresponding  elementary  triangles  may  be  main- 
tained, the  ratio  of  increase  of  each  partial  difference  of  lati- 
tude must  be  the  ratio  of  expansion  of  each  partial  departure, 
and  the  true  difference  of  latitude  CB  (Fig.  78)  be  repre- 
sented by  CB  on  the  chart  (Fig.  79). 

The  triangles  of  Mercator  and  plane  sailing. — The  parts 
involved  in  Mercator  sailing  may  be 
represented  by  a  right  triangle  CEF, 
CE  being  the  augmented  difference  of 
latitude  m,  representing  the  true  differ- 
ence of  Lat.  C  A  =  /;  if  AB  is  drawn 
parallel  'to.EF,  ABC  will  be  the  tri- 
angle of  plane  sailing,  AB  the  de- 
parture, and  CB  the  true  distance  of 
which  the  expansion  on  the  Mercator 
chart  is  CF,  since  the  ratio  between  / 
FIG.  80.  and  m  is  the  same  as  that  between  p 

and  D.  It  is  thus  seen  that  the  tri- 
angle CEF  furnishes  the  formula  for  converting  departure 
into  difference  of  longitude  without  making  the  false  assump- 
tions of  middle  latitude  sailing. 

From  Fig.  80  all  the  formulae  necessary  for  Mercator  sail- 
ing can  be  deduced. 

From  triangle  CEF,  D  =  m  tan  C. ) 
From  triangle  ABC,  d  —    I  sec  C.  \ 

Various  problems  under  Mercator  sailing  may  be  solved 
by  the  above  formulse,  but  those  of  actual  practice  may  be  said 
to  be: 


MERCATOR  SAILING  257 

Case  I.— Required  the  C  and  d  between  two  places  of 
known  position. 

Formulae :  tan  C  =— -,  d=l  sec  C. 
m 

Case  II. — Required  the  latitude  and  longitude  in,  after 
sailing  a  true  C  and  d  from  a  place  of  known  position. 
Formulas :     I  =  d  cos  C,  L2  =  L±  +  I* 
D  =  m  tan  C,  A2  =  Ax  +  D. 

As  the  results  by  Mercator  sailing  and  by  middle  latitude 
sailing  do  not  differ  sensibly  for  small  distances,  the  use  of 
Mercator  sailing  comes  principally  under  Case  I,  when  the 
two  places  are  far  apart. 

When  not  to  use  Mercator  sailing. — Since  in  Mercator  sail- 
ing the  difference  of  longitude  is  found  from  a  formula  in- 
volving tan  C,  and  tangents  of  angles  near  90°  change  very 
rapidly,  it  is  seen  that  any  error  in  m,  the  meridional  differ- 
ence of  latitude,  is  greatly  increased  as  an  error  in  difference 
of  longitude  when  the  course  approaches  90°  or  270°.  In  such 
cases  use  middle  latitude  sailing. 

Use  of  traverse  table, — Problems  in  Mercator  sailing  can 
be  solved  by  the  traverse  table;  the  difference  of  longitude 
and  meridional  difference  of  latitude,  being  respectively  the 
sides  opposite  and  adjacent  in  a  right  triangle,  should  be 
-looked  for  in  the  dep.  and  diff.  of  lat.  columns,  respectively. 
In  using  this  table  where  long  "distances  are  involved,  the 
quantities  given  may  all  be  reduced  by  a  common  divisor  till 
within  the  limits  of  d,  I,  and  p  as  tabulated,  and  the  results 
afterwards  correspondingly  enlarged.  This,  however,  will  in- 
volve some  error  in  results. 

Graphic  solution  of  problems  in  Mercator  sailing  are  made 
in  every-day  navigation,  when  the  reckoning  is  kept  by  con- 
struction on  the  Mercator  chart  as  fully  explained  in  Art.  31. 


258  NAVIGATION 


Ex.  52.  —  Find  the  true  course  and  distance  by  Mercator 
sailing  from 

L±  =  10°  36'  N  \      L2  =  36°  30'  N 
A1==56    34WJt0A2=15    22  W 

By  computation.  — 

o      /  o      / 

LI  =  10  36  N  Jfj  =     635.4  ^  =56  34  W 

X2  =  36  30  N  Jf2  =  2341.3  22  =15  22  W 

Z  =  25  54  N  m  =  1705.9  J)  =41  12  E 

=  1554'    N  =  2472'  E 

Z  =  1554'  N  ..................................  log    3.19145 

D  =  2472'  E  ............  log     3.39305 

m  =  1705.9   ............  log     3.23195 

G  =  N  55°  23'  28"  E  .....  tan  10.16110  ..........  sec  10.24568 

d  =  2736.1  miles  ..............................  log    8.43713 

A       (  Course,  (7N  =  55°  23'  28". 
*'  (Distance,  2736.1  miles. 

By  inspection.  —  Enter  Table  2.  Turn  to  that  page  where 
will  be  found  the  nearest  coincidence,  D  in  dep.  col.  and  m 
in  diff.  lat.  col.  Now,  by  interpolation, 

D  =  247.2,  m  =  173.05;  0  =  55° 
D  =  247.2,  m  =  166.66;  C  —  56° 


D  =  247.2,  ro  =  170.59;  C  =  55°. 38 
Therefore,  by  inspection,  course  is  (7N  =  55°. 38. 
Now  with  the  course  and  I  =  155.4  find  d. 

For  CN  =  55°,      1=  155.4 ;  d  =  271 

CN  =  56°,      I  —  155.4;  d  =  277.8 
Therefore, for CN  =  55°.38, Z  =  155.4;  (2  =  273.58. 

Having  used  a  divisor  of  10  originally,  the  true  distance 
by  inspection  is  2735.8  miles. 

Of  course,  the  above  interpolation,  recorded  for  illustration, 
is  supposed  to  be  done  mentally. 


MIDDLE  LATITUDE  CORRECTION  259 

Ex.  53.  —  Find  true  CN  and  d  from  Brisbane  to  Acapulco. 

o     /     //  o       /     // 

Li  =  27  27  32  S  Ml  =  1703.7  8         \  =  153  01  48  E 

£2  =  16  49  10  N  M2  =  1017.2  N        22  =    99  55  50  W 

I  =  44  16  43  N  m  =  2720.9  N        D  =  107  02  23  E 

=  2656'.  7  N  =  6422'.37 

I  =  2656.7  ...................................   log  3.42434 

D  —  6422'.37  ............  log    3.80770 

m  =  2720.9  .............  log    3.43471 

C  =  N  67°  02'  24"E  ____  tan  10.37299  ..........  sec  10.40884 

d  =  6810.5  miles  ................  .  .............  log  3.83318 

f  Course  0N  =  67°  02'  24". 
{Distance  6810.5  miles. 

133.  To  find  the  value  of  the  correction  to  the  middle 
latitude.  —  In  middle  latitude  sailing,  it  was  stated  that  the 
formula  D  =  p  sec  L0  =  Z  tan  C  sec  L0  was  not  strictly  cor- 
rect, but  that  it  would  be  correct,  if,  to  L0  ,  was  applied  a  cor- 
rection AL,  such  that  the  formula 

D  =  li&nC  sec  (L0  +  AL) 
would  give  the  same  result  as  D  =  m  tan  C. 

From  these  two  may  be  gotten 


-     =  cos  (L0  +  AL)  =  1  -  2  sin 


AL  =  2  *vr*mr~—L0.  (124) 

Values  of  this  correction  have  been  tabulated  where  the 
arguments  are  the  middle  latitude  and  the  difference  of  lati- 
tude. It  has  already  been  stated  that  for  small  values  of  lf 
it  is  unimportant;  and  that  in  those  cases  where  its  use  might 
be  desirable,  it  would  be  better  to  use  Mercator  sailing. 


260  NAVIGATION 

Examples  in  Mercator  Sailing. 
(By  computation.) 

Ex.  54-  —  Find  true  course  and  distance  from  L±  50°  53'  N., 
A  156°  46'  E.,  to  L2  12°  04'  S.,  A2  77°  14'  W. 

JCourse  CN  =  119°  25'  28". 
[Distance  7688.35  miles. 

Ex.  55.  —  Find  true  course  and  distance  from  Lj  42°  20'  N., 
A!  31°  30'  W.,  to  L2  56°  40'  K.,  A2  20°  40'  W. 

Ans  fCourse  CN  =  25°  59'  04". 
*  (Distance  956.7  miles. 

Ex.  56.  —  Find  true  course  and  distance  from  Lt  45°  02'  S., 
A±  20°  19'  W.,  to  L2  65°  20'  S.,  A2  18°  37'  W. 

Ans  JCourse  CN  ~  177°  19'  46"' 
[Distance  1219.4  miles.   . 

Ex.  57.—  A.  ship  sails  from  Lat.  15°  20'  K,  Long.  24°  20' 
W.,  135°  (true)  a  distance  of  2500  miles.  Find  Lat.  and 
Long.  in. 

,       f£2  =  14°  07'48"S. 
1*4=    5     15    48  E. 

Ex.  58.  —  Find  the  true  course  and  distance  by  Mercator 
sailing  from  a  point  in  Lat.  35°  30'  K,  Long.  140°  52'  E. 
(off  Cape  Inaboye,  Japan),  to  a  point  in  Lat.  33°  S.,  Long. 
71°  49'  W.  (off  Valparaiso).  See  Ex.  63  and  Plate  V. 

Course  ^N  — 


Ans 

Distance  9300.55  miles. 


DAY'S  WORK  BY  D.  E. 

134.  In  most  works  on  navigation,  the  subject  of  "Day's 
Work  "  follows  the  sailings,  and  is  considered  without  refer- 
ence to  positions  by  observation;  as  these  are  an  essential 
part  of  the  data  used  in  the  daily  work  of  a  navigator,  this 


DAY'S  WORK  BY  D.  E.  261 

general  subject  is  reserved  till  after  the  chapters  on  latitude 
and  longitude  by  observations  have  been  studied  and  under- 
stood. 

However,  it  must  be  recalled  that  all  the  calculations  enter- 
ing into  the  daily  dead  reckoning  itself  have  been  made, 
and  the  methods  used  have  been  treated,  under  the  head  of 
pilotage,  or,  of  the  several  sailings.  Such  are  the  various 
methods  of  fixing  the  ship's  position  near  land  after  leaving 
port;  taking  departure;  use  of  departure  course  and  distance 
as  a  course  and  distance  of  the  traverse;  correction  for  lee- 
way, variation,  and  deviation,  of  the  various  courses  indicated 
in  the  ship's  log  book;  entry  of  the  true  courses  and  the  dis- 
tances sailed  on  each  in  the  proper  columns  of  the  tabulated 
form ;  consideration  of  the  set  and  drift  of  a  known  current 
as  a  separate  course  and  distance  of  the  traverse;  finding  the 
resultant  difference  of  latitude  and  departure;  the  resultant 
course  and  distance;  conversion  of  departure  into  difference 
of  longitude;  finding  by  D.  E.  the  latitude  and  longitude  at 
end  of  run,  and  the  course  and  distance  to  port  of  destination. 

It  has  been  shown  that  the  noon  position  by  observation  is 
the  true  place  from  which  to  begin  the  dead  reckoning  of  the 
following  day ;  and,  in  case  of  a  discrepancy  between  it  and  the 
position  by  D.  E.,  that  this  discrepancy,  if  not  due  to  inci- 
dental errors  of  navigation,  may  be  attributed  to  current,  the 
set  and  drift  of  which  correspond  to  the  course  and  distance 
from  the  noon  position  by  D.  E.  to  that  by  observation. 

For  the  solution  of  a  day's  work  in  which  positions  by  obser- 
vation are  used,  see  Chapter  XXI. 

In  the  following  example,  a  day's  work  by  D.  E.  is  illus- 
trated. 


262 


NAVIGATION 


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CHAPTEK  VII. 

GREAT  CIRCLE  SAILING.— COMPOSITE  SAILING.— 
GRAPHIC  METHODS. 

135.  Great  circle  sailing  is  the  method  of  solving  problems 
of  navigation  that  arise  from  a  ship  following  a  great  circle 
track  from  point  of  departure  to  destination.  The  arc  of  the 
great  circle  passing  through  two  places  is  the  shortest  dis- 
tance between  these  places,  so  that  a  rhumb  line  is  always 
longer  than  the  great  circle  distance,  except  when  it  coincides 
with  a  meridian  or  the  equator. 

The  rhumb  line  has  long  been  used  by  navigators  because 
of  the  constancy  of  the  course,  the  ease  with  which  it  can  be 
laid  down  on  or  taken  from  a  Mercator  chart,  and  the  sim- 
plicity of  the  calculations  it  involves.  However,  steam  vessels, 
unlike  sailing  vessels,  are  independent  of  winds  and  currents, 
and  are  capable  of  following  that  route  which  means  a  saving 
of  distance  and  of  time;  so  it  is  fair  to  presume  that  in  the 
future  the  great  circle  will  be  followed  as  closely  as  possible 
unless  it  goes  into  latitudes  too  high,  meets  lands,  or  passes 
through  regions  of  ice  and  dangerous  navigation. 

Comparison  of  tracks. — The  difference  in  distance  and, 
hence,  the  saving  of  time  is  less  when  the  loxodrome  ap- 
proaches a  great  circle,  as  is  the  case  between  places  near  the 
equator,  or  near  the  same  meridian ;  the  contrary  holds,  how- 
ever, for  places  in  high  latitudes,  especially  when  differing 
much  in  longitude;  a  case  most  remarkable  for  saving  of  dis- 
tance and  great  divergence  between  the  two  tracks  occurs 
when  the  two  places  are  on  the  same  parallel,  but  differ  180° 
in  longitude.  The  Mercator  course  is  either  East  or  West, 


264  NAVIGATION 

while  the  great  circle  course  is  North  or  South,  across  the  ele- 
vated pole  and  90°  away  from  the  former. 

A  rhumb  line  laid  down  on  a  Mercator  chart  passes  directly 
through  the  point  of  destination ;  the  great  circle  track  plotted 
on  the  same  chart  will  be  a  circuitous  path,  nearer  the  pole 
than  the  Mercator  track,  and  often  going  into  higher  lati- 
tudes than  is  practicable  for  safe  navigation.  The  great 
circle  track  between  two  places  in  different  hemispheres  has 
a  double  curvature  when  plotted  on  the  Mercator  chart,  the 
curve  in  each  hemisphere  being  the  same  in  its  entirety. 
However,  a  ship  following  the  rhumb  line,  steers  the  same 
course,  makes  the  same  angle  with  each  successive  meridian, 
but  never  heads  directly  for  the  port  till  it  is  in  sight,  or  till 
the  end  of  the  voyage ;  another  ship  following  the  great  circle 
track  always  heads  for  the  port,  but  does  so  by  steering  a 
constantly  changing  course. 

Definitions. — The  great  circle  course  is  the  angle  which 
the  great  circle  passing  through  a  place  makes  with  the  me- 
ridian of  that  place.  The  initial  course  is  the  angle  which 
the  great  circle  through  points  of  departure  and  destination 
makes  with  the  meridian  of  point  of  departure;  the  final 
course  is  the  angle  which  it  makes  with  the  meridian  of  desti- 
nation. 

The  distance  is  the  length  of  the  arc  of  the  great  circle 
which  forms  the  path  of  the  ship  between  the  two  points,  ex- 
pressed in  nautical  miles. 

Vertices. — In  accordance  with  geometrical  principles,  the 
equator  bisects  the  great  circle  passing  through  the  two 
places,  and  that  point  of  the  circle  in  each  hemisphere  which 
is  farthest  from  the  equator  is  the  vertex  in  that  hemisphere ; 
in  other  words,  the  vertices  are  the  points  of  highest  latitude. 

Only  one  vertex  is  considered,  and  that  is  in  the  hemi- 
sphere whose  pole  determines  the  course.  The  vertex  .may  or 
may  not  be  between  the  two  places.  If  the  initial  and  final 


GREAT  CIRCLE  SAILING  265 

courses  are  both  less  than  90°,  the  vertex  falls  between  the 
two  places;  if  one  is  greater  than  90°,  the  vertex  falls  on  the 
arc  produced  180°  from  this  course. 

Points  of  maximum  separation. — Since  all  points  of  the 
great  circle  track  betwen  two  places  in  the  same  hemisphere, 
except  those  of  departure  and  destination,  are  nearer  the  pole 
than  the  rhumb  line,  it  follows  that  there  must  be  some  one 
point  where  the  meridian  distance  between  the  two  tracks  is 
greatest,  and  this  is  the  point  of  maximum  separation.  It  is 
apparent  that  at  this  point  the  courses  on  the  two  tracks  are 
equal.  Hence,  knowing  the  rhumb  course,  it  is  only  neces- 
sary to  find  that  point  of  the  arc  where  the  great  circle  course 
would  equal  it. 

Finding  great  circle  course  and  distance. — There  are  four 
general  methods  for  solving  the  great  circle  problem : 

(1)  By  computation. 

(2)  By  azimuth  tables. 

(3)  By  great  circle  charts. 

(4)  By  graphic  approximation. 

By  computation. — The  problem  consists  in  the  solution  by 
spherical  trigonometry  of  a  spherical  triangle,  formed  by  the 
meridians  passing  through  the  two  places  and  the  great  circle 
arc  forming  the  ship's  track.  The  lengths  of  the  two  sides 
are  known,  being  equal  to  the  co.  latitudes  of  the  two  places; 
the  included  angle  at  the  pole  is  the  difference  of  longitude 
of  the  two  places;  so  the  triangle,  having  two  sides  and  the 
included  angle  given,  may  be  solved  by  Napier's  Analogies, 
but  preferably  by  Napier's  Eules. 

To  plot  the  curve,  not  only  the  vertex,  but  a  series  of 
points  along  the  curve  should  be  determined  by  their  coordi- 
nates; and,  having  been  plotted  on  the  Mercator  chart,  the 
curve  may  be  traced  through  them. 

In  Fig.  81,  let  A  be  the  point  of  departure,  B  the  point  of 
destination,  AB  the  great  circle  passing  through  them,  V  its 


266 


NAVIGATION 


vertex,  m1?  ra2,  etc.,  points  along  the  curve,  P  the  elevated  pole. 
Position  of  A,  Lat.  L± ,  Long.  A± ;  therefore,  AP  =  90°  —  £± . 

Position  of  B,  Lat.  L2 ,  Long. 
A2;  therefore,  PB  =  90°— L2 . 
APB  =  difference  of  longi- 
tude of  A  and  B  —  A2  ^  A± . 
(7X  is  the  initial  course ;  C2 , 
the  final  course,  d  =  dis- 
tance A 5.  Drop  a  perpendic- 
ular BO  (=  fc)  on  J.P,  so  that 
one  triangle  P£0  shall  include 
two  of  the  known  parts,  and 
the  triangle  ABO  shall  in- 
elude  the  required  C±  and  6?. 
Also,  dividing  AP  into  two 
parts,  PO  =  <£  and  OA  =  90°  —  (L±  +  </>). 

To  find  the  initial  course  <7±. — Applying  Napier's  Eules  to 
triangle  P05^  cos  (A2  ^  A±)  =  tan  <f>  tan  L2 . 

or,  tan  </>  =  cos  (A2  ^  A±)  cot  L2 ,  (1^5) 

also,  sin  <j>  =  cot  (A2  ^  Ai)  tan  fc;  ) 
to  triangle  AOB,  cos  (L±  +  ^>)  =  cot  ^  tan  A;;  J 
therefore,  cot  0±  =  cot  (A2  ^  A±)  cos  (L±  +  </>)  cosec  </>.  (126) 
To  find  the  distance  d  =  AB. — Proceeding  as  above, 
sin  L2  =  cos  <f>  cos  Tcf 

cos  d  =  sin  (L±  +  <^>)  cos  fc, 

therefore,  cos  6Z  =  sin  (L±  +  </>)  sin  L2  sec  <£  .    (127) 
The  distance  is  found  in  degrees,  minutes,  etc.,  of  a  great 
circle  which  will  be  reduced  to  minutes  for  distance  in  nauti- 
cal miles. 

PV  is  the  arc  of  a  meridian  perpendicular  to  the  great 
circle  track ;  therefore  PVA  and  PVB  are  right  angles. 
V  is  the  vertex ;  Lat.  Lv ,  Long.  Xv  . 

The  vertex  lies  between  A  and  B,  unless  either  (7X  or  (72  is 
>  90°. 


GREAT  CIRCLE  SAILING  267 

To  find  the  latitude  and  longitude  of  the  vertex. — In  the 

right  triangle  APVf  PV  =  90°  —  Lv  and  A'PV  =  \v  ~  A± , 
therefore,  by  Napier's  Rules, 

cos  Lv  =  cos  L-L  sin  (7± ,  (128) 

sin  LI  =  cot  <7±  cot  (A*  ~  A^,   )  ,129) 

cot  (Au  ^  A±)  =  sin  LI  tan  C± .  J 

To  find  the  latitude  and  longitude  6f  other  points  of  the 
curve. — Assume  meridians  Pm± ,  Pm2 ,  etc.,  differing  in  longi- 
tude 019  02 ,  etc.,  (say  5°  or  10°,  if  desired)  from  the  longi- 
tude of  vertex,  and  solve  the  right  triangles  thus  formed  by 
Napier's  Eules. 

Therefore,  tan  Lmi  =  cos  0±  tan  Lv ,    ,      )         (       . 
,       -.-  f\    i       T     etc.    >         ( .LoU) 

tan  Lm2  =  cos  02  tan  Lv ,          j 

Each  of  these  last  formulae  will  give  a  position  each  side 
of  the  vertex,  or  two  points  of  the  curve;  thus 
from  first  equation, 

Lat.  m^ ,  Long.  (Xv  —  0±)  and  Long.  (Au  +  0±), 
from  second  equation, 

Lat.  m2 ,  Long. .  (Xv  —  02)  and  Long.  (At,  +  02) . 

It  must  not  be  forgotten  that  the  course  is  ever  varying, 
and  that  the  course  to  be  steered  at  any  meridian  is  the  angle 
which  that  meridian  makes  with  the  track.  By  the  solution, 
the  angle  will  be  found  from  the  elevated  pole  towards  the 
East  or  West, but, if  it  is  found  to  be  greater  than  90°,  as  when 
the  vertex  is,  or  has  been  left  behind,  it  may  be  convenient  to 
name  the  course  as  from  the  depressed  pole,  or  the  supplement 
of  the  angle  found  by  computation  in  the  triangle  of  which 
the  elevated  pole  is  one  point. 

To  find  the  course  in  any  longitude  A^. — In  Fig.  81,  PVG 
is  a  right  triangle,  PV  =  90°  -  -  Lv ,  VPG  =  A,  ~  A^ . 
Let  Cg  be  the  course  in  Long.  A^ ,  and  Cq  be  the  course  at  the 
equator ;  then,  by  Napier's  Eules, 


268  NAVIGATION 

cos  Cg  —  sin  Lv  sin  (\v  ~  Xg),  (131) 

and  for  the  final  course  C2  in  Lat.  L2 ,  Long.  A2 , 

cos  (72  =  sin  Lt;  sin  (A^  ~  A2).  (132) 

At  the  point  of  crossing  the  equator  sin  (Xv  ~  \q)  =  sin  90°, 
therefore,  at  the  equator, 

Cq  =  Co.Lv,  (133) 

and  Ag  =  ^±90°.  (134) 

Precautions. — In  solving  the  triangle,  let  the  elevated  pole 
(the  pole  of  that  hemisphere  in  which  lies  the  position  L^) 
be  at  one  angle  of  the  triangle,  regard  L^  as  positive  and  L2 , 
when  of  a  different  name  from  L± ,  as  negative.  Strict  regard 
must  be  had  to  the  signs  of  the  functions.  <j>  may  be  taken 
out  as  positive  up  to  180° ;  or,  if  tan  (/>  is  negative,  instead  of 
taking  <j>  in  the  second  quadrant,  it  may  be  regarded  as  nega- 
tive, the  foot  of  the  perpendicular  falling  the  other  side  of  the 
pole.  The  angle  GI  will  be  found  with  its  correct  value,  if 
attention  is  paid  to  the  signs.  The  course  is  from  the  elevated 
pole,  East  or  West,  as  B  is  East  or  West  of  A. 

Cg  and  C2  are  reckoned  from  the  elevated  pole  when  ap- 
proaching the  vertex  and  from  the  depressed  pole  when  going 
away  from  it,  toward  East  or  West  according  as  the  ship  is 
proceeding  eastward  or  westward. 

The  fact  of  the  vertex  being  ahead  or  astern  is  determined 
by  a  comparison  of  its  longitude  with  that  of  the  meridian 
from  which  the  course  is  taken. 

Though,  when  considering  the  above  mentioned  courses  as 
angles  of  a  spherical  triangle,  the  method  given  is  the  proper 
one  to  pursue  in  the  solution  of  that  triangle ;  still,  as  soon  as 
found,  the  course,  for  practical  purposes,  should  be  expressed 
in  the  more  convenient  form  of  CN  which  is  measured  from 
North,  around  to  the  right,  from  0°  to  360°.  CN  will  be 
simply  z°,  180°—  z°,  180°  +  z°,  or  360°—  z°,  according 
as  the  course  by  solution  is  N.  x°  E.,  S.  x°  E.,  S.  x°  W.,  or 
N.  x°  W.,  respectively. 


PLATE  IV, 


270 


NAVIGATION 


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FORM  AND  EXAMPLE 


271 


Ex.  61.  —  In  the  above  example  find  the  coordinates  of  two 
points  of  the  curve,  one  in  a  longitude  40°  to  eastward  of  the 
vertex,  the  other  40°  to  the  westward.  Find  also  the  course 
at  this  western  point  and  the  final  course  (see  Fig.  81). 


Here  0  =  40°,  \»  ~  \d  —  40 


X2  =  82°  37'  18", 


Formulas 


tan  Lm  =  tan  Lv  cos  0. 


cos 


=  82° 

37' 

0 

=  40 

=  72 

34 

—  67 

43 

N 

52 

10 

N 

18 

52 

?,  -s 
\  =s 

in  Lv  sin  (A^  ^ 
in  LU  sin  (A^  '• 

-A,). 
-A2). 

18" 

sin  9.99639 

cos    9.88425 

sin  9.80807 

20  S 

tan  10.50319 

sin  9.97959 

sin  9.97959 

00  S 

tan  10.38744 

22  W 

cos  9.78766 

54  W 

cos  9.97598 

«  . 

COOrdmate6o£P°lnt3 


East  of  vertex     Lat.  67°  43'  00"  S.,    Long.  81°  07'  18"  E. 


West  of  vertex  Lat.  67    43   00     S.,    Long. 
Course  at  Western  Point  <7N  =  307°  49'  38". 
Final  conrse  (7N  =  341°  07'  06". 


1    07   18    E. 


272 


NAVIGATION 


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FORM  AND  EXAMPLE 


273 


To'  find  the  point  of  maximum  separation. — The  point  of 
maximum  separation  between  the  great  circle  and  Mercator 
track  is  the  point  where  the  two  courses  are  the  same. 


FIG.  82. 


In  Fig.  82,  A  B  is  the  Mercator  track,  A  SB  the  great  circle 
route,  on  which  V  is  the  vertex  and  8  the  point  of  maximum 
separation.  To  find  this  latter  point  it  is  only  necessary  to 
solve  the  right  triangle  VPS  for  PS  (=  90°  —  Lm8)  and 
Z  VPS  (=  \v  ~  Ams),  where  Lms  is  the  Lat.  and  ATOS  the 
longitude  of  the  point,  having  given  PV  =  90°  —  Lv  and 
£  PSV  —  the  Mercator  course.  The  point  is  one  of  but  little 
practical  value. 


PLATE  V. 


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FORM  AND  EXAMPLE 


275 


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276  NAVIGATION 

(b)  To  find  the  points  of  maximum  separation,  find  the 
coordinates  of  the  point  of  the  curve  in  each  hemisphere 
where  the  great  circle  course  equals  the  Mercator  course 
46'  28". 


Lv  ............  —    35°  43'  07"  ......  cos.    9.90950  ......  cosec.  10.23373 

Her.  C  ........  =863    46   28  W  ..  .cosec  10.04718  ........  cos    9.64533 

Aw  ...........   =    25    10  ..........  cos    9.95668 

A,—  a*.  .......  =    49    11    45  ...........................  sin    9.87906 

*,  in  North  Lat.  =  133°  36'  25"  E  \  in  South  Lat.  .  .  =46°  23'  35"  W 


49     11   45  ~         .........  =  49    11    45 


JU,.  in  North  Lat.=  177    11   50  W     1  j  ^  in  South  Lat.     =  95    35    20  W 
Lat.. =    25    ION  J  1  Lat =25    10  S 

(c)   The  results  of  the  solution  of  this  subdivision  of  the 
example  will  be  found  tabulated  on  the  opposite  page. 

Finding  the  great  circle  course  from  azimuth  tables. — The 

azimuth  of  a  heavenly  body  is  tabulated  in  the  tables  with  the 
arguments  L,  d,  and  t.  Now,  from  the  observer's  position  the 
azimuth  of  a  heavenly  body  is  the  same  as  the  great  circle 
course  to  that  terrestrial  position  having  the  same  heavenly 
body  in  its  zenith ;  so,  to  use  the  azimuth  tables  for  finding  the 
great  circle  course  to  a  place,  it  is  only  necessary  to  substitute 
for  the  body's  declination  the  latitude  of  destination,  for  the 
body's  hour  angle  the  difference  of  longitude  between  the 
places  expressed  in  time,  and  to  consider  the  latitude  of  de- 
parture as  that  of  the  observer.  The  rules  for  marking  the 
azimuth  apply  for  marking  the  course. 

Solution  by  gnomonic  charts,  or  great  circle  sailing  charts. 

— This  subject  has  already  been  considered  under  the  head  of 
gnomonic  charts  (Art.  24) .  Especial  reference  is  made  to  the 
gnomonic  charts  issued  by  the  IT.  S.  Hydrographic  Office,  on 
which  are  provided  the  means  of  determining  the  great  circle 


FORM  AND  EXAMPLE 


277 


£  - 


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FH    ^ 


278  NAVIGATION 

course  and  distance  directly,  without  transferring  positions 
to  a  Mercator  chart.  Eeference  is  also  made  to  the  polar 
chart  (Art.  25),  which  is  available  for  either  hemisphere.  Any 
meridian  may  be  taken  as  that  of  Greenwich,  and  the  two 
places,  between  which  the  great  circle  course  and  distance  are 
desired,  having  been  plotted,  join  them  by  a  straight  line. 
This  line  is  the  great  circle  track  from  which  any  number 
of  coordinates  may  be  transferred  to  a  Mercator  chart.  The 
polar  chart  is  especially  available  in  the  Southern  hemisphere 
where  great  circle  sailing  possesses  so  many  advantages. 

Solution  by  graphic  methods;  Use  of  terrestrial  globe. — 
Locate  the  two  places  on  the  globe,  move  it  till  both  places 
coincide  with  the  upper  edge  of  the  horizon  circle.  Draw  a 
line  between  the  two  points  along  the  edge  of  the  horizon 
circle.  This  will  be  the  required  great  circle  distance  which 
can  be  measured  by  the  scale  on  the  horizon  circle,  and,  when 
reduced  to  minutes  of  arc,  will  be  the  distance  in  nautical 
miles.  Take  off  the  latitudes  and  longitudes  of  as  many 
points  as  may  be  desired,  transfer  them  to  a  Mercator  chart, 
and  trace  in  the  arc.  The  courses  and  distances  from  point 
to  point  on  this  arc  may  be  gotten  directly  from  the  chart ;  or, 
by  computation,  using  middle  latitude  or  Mercator  sailing. 

Graphic  chart  methods. — Various  methods  are  now  used  to 
lay  down  a  great  circle  track  on  a  Mercator  chart.  These  ob- 
viate the  calculations  which,  by  some  people,  may  be  consid- 
ered laborious. 

Towson's  method  permits  a  track  to  be  laid  down  on  a  Mer- 
cator chart  with  a  great  degree  of  accuracy.  His  linear  index 
gives  the  latitude  and  longitude  of  the  vertex,  whilst  the 
accompanying  tables  give  the  true  course  at  every  degree  of 
longitude  from  the  vertex. 

Airy's  method. — The  following  method,  proposed  by  Profes- 
sor Airy,  when  Astronomer  Eoyal,  lays  down  a  curve  which  is 
a  very  close  approximation  to  the  great  circle  arc : 


SOLUTION  BY  GRAPHIC  METHODS 


279 


(1)  Join  the  two  places  on  the  chart  by  a  straight  line. 
Erect  a  perpendicular  at  its  middle  point,  on  the  side  next  to 
the  equator,  producing  the  perpendicular  beyond  the  equator, 
if  necessary. 

(2)  Find  the  middle  latitude  between  the  two  places,  and 
with  this  middle  latitude  enter  the  table  below  and  take  out 
the  corresponding  parallel.     The  intersection  of  this  parallel 
with  the  perpendicular  will  be  the  center  of  the  required  arc. 


Middle 
Latitude. 

Name. 

Corresponding 
Parallel. 

Middle 
Latitude. 

Name. 

Corresponding1 
Parallel. 

0 

0 

0 

0       ' 

20 

Opposite 

81  13 

52 

Opposite 

11  33 

22 

78  16 

54 

6  24 

24 

74  59 

56 

1  13 

26 

71  26 

58 

Same 

4     0 

28 

67  38 

60 

9  15 

30 

63  37 

62 

14  32 

32 

59  25 

64 

19  50 

34 

55  05 

66 

25  09 

36 

50  36 

68 

30  30 

38 

46     0 

70 

35   52 

40 

41   18 

72 

41  14 

42 

36  31 

74 

46  37 

44 

31  38 

76 

52     1 

46 

26  42 

78 

57  25 

48 

21.  42 

80 

62  51 

50 

16  39 

An  approximate  great  circle  track  may  be  thus  laid  down : 
Compute  the  initial  and  final  great  circle  courses  between  the 
two  places  A .  and  B.  Join  AB  on  the  chart,  erect  a  perpen- 
dicular at  its  middle  point.  Find  the  differences  between  the 
Mercator  and  the  two  computed  great  circle  courses.  Lay  off 
the  angles  DAB  and  EAB  equal  to  these  differences.  Erect 
perpendiculars  to  AD  and  AEf  cutting  the  first  perpendicular 
in  p'  and  p.  The  point  c  midway  between  p  and  p'  will  be 
the  center  of  the  required  arc  whose  radius  will  be  cA  (Fig. 
83). 


280  NAVIGATION 

136.  Composite  sailing. — Whenever  the  great  circle  track 
passes  into  higher  latitudes  than  it  is  practicable  or  desirable 
to  go,  some  of  the  advantages  may  be  secured  without  going 
into  regions  of  ice  and  danger,  by 
following  a  composite  track,  a  form 
of  sailing  first  proposed  by  Mr.  Tow- 
son.  Decide  on  the  parallel  above 
which  it  is  inadvisable  to  go,  sail  on 
the  arc  of  a  great  circle  which  passes 
through  the  point  of  departure  and 
has  its  vertex  on  this  limiting  paral- 
lel, proceed  along  the  parallel  till 
there  is  met  a  second  great  circle 
which,  passing  through  the  point  of 

destination,  has  its  vertex  also  on  the  limiting  parallel;  then 
follow  this  arc  to  destination. 

There  are  three  general  methods  used  in  composite  sailing : 
(1)  By  gnomonic  charts.  (2)  By  computation.  (3)  By 
graphic  methods. — 

By  gnomonic  charts. — Draw  lines  from  points  of  departure 
and  destination  tangent  to  the  limiting  parallel.  In  the  case 
of  the  great  circle  sailing  charts  of  tfie  U.  S.  Hydrographic 
Office,  find  the  track  from  point  of  dopjirture  to  point  of 
tangency  and  from  second  point  of  tangency  to  point  of  des- 
tination, the  intervening  distance  being  found  along  the  par- 
allel from  a  Mercator  chart  or  by  parallel  sailing.  On  a  polar 
chart  (see  Art.  25)  tangents  are  drawn  in  the  same  way  to 
the  limiting  parallel.  Suppose  it  is  desired  to  find  the  com- 
posite track  from  L±  =  45°  N.,  A±  =  150°  E.,  to  L2  =  47J° 
N.,  X2  =  130°  W.,  the  limiting  parallel  being  50°  N.  From 
the  first  position  draw  CE  (Fig.  10)  tangent  to  the  parallel 
of  50°,  and  DF  tangent  to  the  same  parallel;  C  and  D  being, 
respectively,  the  points  of  departure  and  destination.  Transfer 
any  desired  number  of  points,  including  points  of  tangency, 


COMPOSITE  SAILING 


281 


from  the  track  on  the  gnomonic  chart  to  the  Mercator  chart, 
by  the  coordinates  of  latitude  and  longitude.  Then  sail  from 
point  to  point  of  the  first  great  circle  till  the  parallel  is 
reached,  along  the  parallel,  and  from  point  to  point  of  the 
second  great  circle  to  destination. 


FIG.  84. 


By  computation.  —  In  Fig.  84,  let  BE  be  the  limiting  par- 
allel, AB  and  EF  the  great  circles  which  pass,  respectively, 
through  the  points  of  departure  A  and  destination  F,  and 
have  their  vertices  on  the  limiting  parallel.  The  composite 
track  will  be  ABEF.  Since  the  limiting  parallel  furnishes 
the  Lv  in  each  case,  PB  =  PE  =  CoLv  .  Letting  the  differ- 
ent elements  that  enter  be  represented  as  indicated  in  Fig.  84, 
by  Napier's  rules,  we  have, 

In  triangle  ABP,  sin  (7±  =  cos  Lv  sec  L^. 
cos  D±  =  cot  Lv  tan  L±. 
cos  d:  =  cosec  Lv  sin  L±. 
In  triangle  FEP,  sin  C2  =  cos  Lv  sec  L2. 

cos  D2  =  cot  Lv  tan  L2.  -(135) 

cos  d2  =  cosec  Lv  sin  L2< 
In  triangle  BEP,  D  =  (\2  ~  AJ  —  (Z^  +  D2), 

p  =  D  cos  Lv. 
Composite  distance  =  d  =  d-^  +  p  +  d2. 


282  NAVIGATION 

Ex.64. — From  example  60  (Art.  135), we  have  L±  35°  40' S., 
A!  118°  06'  07"  E.,  L2  22°  15'  S.,  A2  41°  30'  W. ;  it  is  required 
to  find  the  shortest  possible  route  by  composite  sailing  when 
the  60th  parallel  of  South  Lat.  is  the  limiting  parallel. 

Find  the  distance,  initial  and  final  courses. 

Solution  of  the  eastern  triangle  Fig.  84. 

O          /          It 

LI      =     3540008  sec  10.09022          tan  9.85594          sin      9.76572 

Lv      =     60  S  cos    9.69897          cot  9.76144          cosec  0.06247 


8  37  59  03  W          sin    9.78919 


=     65  31  15  cos  9.61738 


d1       =     47  40  48— 2860'. 8 cos  9.82819 

Solution  of  the  western  triangle  Fig.  84. 

o  t  n 

£2  =  2215    S     sec  10.03360     tan  9.61184     sin    9.57824 
Lv  =  60      S    cos  9.69897     cot  9.76144     cosec  10.06247 

sin  9.73257 


....  cos  9.37328 

d^—  64  04  20  —  3844'.33 cos         9.64071 

D  =  aa  ~  3,,)  -  (A  +£2)  D  =  1064.62  log  3.02719 

D  =  17°  44'  37"  Lv  60°  cos  9. 69897 

D—  1064'.62  p    ~     532.3  log  2.72616 

Initial  course  <7N  =  217°  59'  03"  dt  —  2860.8 

Final  course   (7N  =  327°  18'  06"  dy  —  3844.33    - 

See  plate  (IV).  d  —  p  +  dl  +  d^  —  7237.43  miles. 

Comparing  the  great  circle  and  composite  distances  in  this 
example,  it  is  seen  that  the  great  circle  distance  is  7136.86 
miles,  the  composite  distance  7237.43  miles,  or  that  there  is  a 
difference  of  100.57  miles  in  favor  of  the  great  circle. 

Graphic  methods. — In  graphic  methods,  use  may  be  made 
of  the  terrestrial  globe,  or  the  track  may  be  laid  down  on  a 
Mercator  chart  approximately  as  follows : 


GRAPHIC  METHODS  283 

Decide  on  a  limiting  parallel.  Join  the  two  places  on  the 
chart  by  a  straight  line,  at  whose  center  erect  a  perpendicular 
and  prolong  it  till  it  meets  the  limiting  parallel.  Through  this 
point  of  intersection  and  the  two  given  points  pass  a  circle; 
then  sail  from  point  to  point  on  this  circular  route,  by  middle 
latitude  or  Mercator  sailing,  till  the  limiting  parallel  is 
reached;  along  that  parallel  to  the  second  point  of  intersec- 
tion with  the  circle ;  then  from  point  to  point  of  the  remainder 
of  the  circle,  by  middle  latitude  or  Mercator  sailing,  till  des- 
tination is  arrived  at. 

Examples  Under  Great  Circle  Sailing. 

Ex.  65. — Find  the  great  circle  initial  course  and  distance 
from  Melbourne  in  Lat.  37°  49'  53"  S.,  Long.  144°  58'  42"  E., 
to  Callao  in  Lat.  12°  03'  53"  S.,  Long.  77°  08'  20"  W.  Also 
Lat.  and  Long,  of  the  vertex. 

N  =  132°  55'  36". 
,   d  =  6984.37  miles. 
'*  Lv  =  54°  40'  00"  S. 
A*  =  158°  25' 27"  W. 

Ex.  66. — Find  the  great  circle  initial  course  and  distance 
from  San  Francisco  in  Lat.  37°  47'  30"  N.,  Long.  122°  27'  49" 
W.,  to  Sydney  in  Lat.  33°  51'  41"  S.,  Long.  151°  12'  39"  E. 
Also  position  of  vertex. 

•<7N=r  240°  17'  10". 
.       ,   d  —  6445.25  miles. 
.£,=    46°  39' 32"  S. 

\v  =  100°  30'  oi"  E; 

Ex.  67. — (a)  Find  the  great  circle  initial  course  and  dis- 
tance from  Cape  Vanderlind,  Urup  I.,  Lat.  45°  37'  N.,  Long. 
149°  34'  E.,,  to  Pt.  Eeyes  Lt.  Ho.  on  the  coast  of  California, 
Lat.  37°  59'  39"  K,  Long.  123°  01'  24"  W.  Also  Lat.  and 
Long,  of  vertex. 


284  NAVIGATION 

(&)  Find  the  longitudes  of  intersection  of  the  great  circle 
with  the  50th  parallel  of  North  latitude,  and  the  course  at  first 
intersection. 

(c)  Not  wishing  to  go  further  North  than  the  50th  parallel, 
on  account  of  the  Aleutian  Islands,  find  the  increase  of  dis- 
tance by  pursuing  the  50th  parallel  from  the  1st  to  the  2d 
intersection,  instead  of  following  the  great  circle  entirely. 
'0N  =62°  46'  15". 
^  =  3738  miles. 
}^LV=    51°  32' 26"  N. 
A,,  =  174°  40' 45"  W. 

:Long.  of  West  intersection,  166°  30'  46"  E. 
Long,  of  East  intersection,  155°  52'  16"  W. 
Course  at  West  intersection,^  =  75°  22'  19". 
(c)  Increase  of  distance,  14.7  miles. 

Ex.  68. — Find  the  great  circle  initial  course  and  distance 
from  Brisbane,  Australia,  Lat.  27°  27'  32"  S.,  Long.  153°  01' 
48"  E.,  to  Acapulco,  Lat.  16°  49'  10"  N.,  Long.  99°  55'  50"  W. 
Also  Lat.  and  Long,  of  the  vertex. 

J/N=82°  04' 28". 
d  =  6748.63  miles. 
Lv=    28 °  29' 44"  S. 
=  136°  13'45"E. 
Ex.  69. — Find  the  great  circle  initial  course  and  distance 
from  a  point  off  Cape  Agulhas  in  Lat.  34°  55'  S.,  Long.  20° 
01'  E.,  to  a  point  off  Java  Head  in  Lat.  6°  55'  S.,  Long.  105° 
02'  E.     Also  Lat.  and  Long,  of  the  vertex. 

fC'N=92°  51'  32". 
d  =  4918.4  miles. 


AU  =  25°00'11"E. 


PART  II. 

NAUTICAL  ASTRONOMY. 


CHAPTER  VIII. 

GENERAL  DEFINITIONS.— THE  VARIOUS  SYSTEMS  OF 

SPHERICAL  CO-ORDINATES,  AND  CORRELATED 

TERMS. 

137.  Nautical  astronomy  is  a  special  application  of  practi- 
cal astronomy  to  the  needs  of  seagoing  people,  who,  by  obser- 
vations of  the  heavenly  bodies,  are  enabled  to  determine  the 
latitude  and  longitude  at  sea,  and  the  error  of  their  principal 
navigational  instruments,  the  chronometer,  the  compass,  and 
the  sextant. 

The  heavenly  bodies  are  the  fixed  stars,  and  those  bodies 
constituting  what  is  known  as  the  solar  system;  namely,  the 
sun,  the  planets  and  their  satellites,  comets,  and  meteors.  The 
fixed  stars,  numbering  many  millions,  are  situated  at  immense 
distances  beyond  the  limits  of  the  solar  system.  Of  all  these 
heavenly  bodies,  only  the  following  need  be  considered  for 
navigational  purposes :  the  sun,  the  moon,  four  planets  (Mars, 
Venus,  Jupiter,  and  Saturn),  and  about  30  fixed  stars. 

Astronomy  teaches  that  the  planets  revolve  about  the  sun, 
from  West  to  East  in  elliptical  orbits,  at  varying  rates  of 
speed,  according  to  their  positions  in  their  orbits  as  well  as 
their  distances  from  the  sun,  and  at  the  same  time  rotate  on 
their-  axes.  The  period  of  a  complete  revolution,  or  time 
required  to  move  through  360°  in  its  orbit,  constitutes  the 
planet's  sidereal  period  or  year ;  and  the  period  of  a  complete 
rotation  on  its  axis  is  a  planet's  sidereal  day. 

The  earth  is  one  of  the  planets  of  the  solar  system ;  its  orbit 
is  in  a  plane  inclined  about  23°  27 y  to  the  plane  of  the  equi- 


288 


NAUTICAL  ASTRONOMY 


noctial.  The  form  of  this  orbit  is  elliptical,  the  sun  being  at 
one  of  the  foci.  The  advance  of  the  earth  in  its  orbit  is  irreg- 
ular, being  the  most  rapid  near  perihelion,  about  January  1, 
and  slowest  near  aphelion,  about  July  1. 


138.  The  celestial  sphere. — To  an  observer  on  the  earth's 
surface  all  the  heavenly  bodies  appear  to  lie  upon  the  concave 
surface  of  a  sphere  of  indefinite  radius  of  which  only  half  is 
visible,  the  other  half  being  cut  off  by  the  horizon.  Owing  to 
the  insignificant  ratio  of  the  earth's  radius  to  that  of  this 
assumed  sphere,  the  eye  of  the  observer  may  be  considered  as 
being  at  0,  the  earth's  center.  This  sphere  is  called  the  celes- 
tial sphere.  However,  these  bodies,  like  r,  i,  u,  v,  w,  etc.  (Fig. 
85),  are  not  at  the  same  distance  from  the  observer,  and, 
being  projected  on  the  celestial  concave,  their  apparent  posi- 
tions depend  on  their  directions  only  and  not  on  linear  dis- 


GENERAL  DEFINITIONS 


289 


tances.  In  like  manner,  the  various  fixed  points  and  circles 
of  the  terrestrial  globe  defined  in  Chapter  I  may  be  projected 
on  the  celestial  sphere,  the  point  of  sight  considered  to  be  at 
the  center  of  the  earth,  0  in  Fig.  85. 

The  axis  of  the  celestial  sphere  is  the  indefinite  prolonga- 
tion of  the  earth's  axis  intersecting  the  celestial  sphere  in  two 
points  called  the  celestial  poles,  corresponding  to  and  named 
like  the  North  and  South  poles  of  the  earth.  That  pole  above 


the  horizon  at  any  place  is  known  as  the  elevated  pole,  the  one 
below  the  horizon  as  the  depressed  pole.  Axis  PP'  (Figs.  85 
and  86). 

The  celestial  equater,  also  called  the  equinoctial,  is  the 
great  circle  of  the  celestial  sphere  in  which  the  plane  of  the 
terrestrial  equator,  indefinitely  extended,  meets  the  celestial 
sphere;  EQ  (Fig.  85),  EDWC  (Fig.  86). 

Horizons. — A  plane  passed  tangent  to  the  earth's  surface  at 
the  feet  of  the  observer  will  be  his  sensible  horizon,  H'Hf  (Fig. 
85) ;  a  second  plane  parallel  to  this  through  the  center  of  the 


290 


NAUTICAL  ASTRONOMY 


earth  will  be  his  rational  horizon,  N8  (Fig.  85) ;  and  these  two 
planes  indefinitely  extended  intersect  the  celestial  sphere  in 
practically  one  great  circle  called  the  celestial  horizon.  The 
point  Z  directly  over  the  observer's  head  is  the  Zenith,  the 
point  Na,  directly  opposite  and  under  his  feet,  is  the  Nadir. 

The  celestial  meridian  is  the  great  circle  of  the  celestial 
sphere  passing  through  the  poles  of  the  heavens,  the  zenith 
and  nadir.  It  intersects  the  horizon  in  the  North  and  South 
points,  the  North  point  being  the  one  nearer  the  North  pole. 
It  is  the  great  circle  of  the  celestial  sphere  cut  out  by  the 
indefinite  extension  of  the  plane  of  the  terrestrial  meridian. 
That  semicircle  which  lies  on  the  same  side  of  the  axis  as  the 
zenith  is  the  upper  branch;  the  other  semicircle  is  the  lower 
branch  of  the  meridian.  In  Fig.  85,  PQP'E  is  the  meridian 
(in  this  particular  figure  it  is  also  the  solstitial  colure),  PQP' 
is  the  upper  branch,  PEP'  the  lower  branch  of  the  meridian. 

In  Fig.  86,  PZP'D  is  the  me- 
ridian, PZP'  being  the  upper 
branch. 

Ecliptic. — Though  the  earth 
in  reality  moves  around  the 
sun,  completing  its  revolution 
of  360°  in  one  sidereal  year, 
the  sun's  center  apparently  de- 
scribes a  circle  in  the  opposite 
direction  on  the  celestial 
sphere,  and  this  great  circle  is 
the  ecliptic,  CC'  (Fig.  85) ;  also  OTC".(Fig.  87),  a  projection 
on  the  plane  of  the  horizon. 

Angle  of  planes  of  ecliptic  and  celestial  equator. — The 

celestial  equator  is  inclined  to  the  ecliptic  at  the  same  angle 

that  the  earth's  equator  is  inclined  to  the  earth's  orbit,  about 

23°  2Vy. 

Equinoctial  and  ecliptic  points. — The  two  opposite  points 


FIG.  87. 


GENERAL  DEFINITIONS  291 

of  intersection  of  the  equinoctial  and  ecliptic  are  practically 
fixed  points  on  the  celestial  sphere. 

The  sun's  center  crosses  the  equator  twice  a  year,  once 
about  March  21,  once  about  September  21,  and  these  being 
times  of  equal  day  and  night,  are  called  equinoxes,  and  the 
points  of  crossing,  equinoctial  points.  The  point  known 
as  the  first  point  of  Aries  is  the  point  of  the  equinoctial  occu- 
pied by  the  sun  in  passing  from  the  southern  to  the  northern 
hemisphere,  on  or  about  March  21 ;  hence  it  is  called  the  vernal 
equinoctial  point.  The  other  point  is  occupied  by  the  sun's 
center  on  or  about  September  21,  and  is  called  the  autumnal 
equinoctial  point.  Though  now  about  30°  distant  respectively 
from  the  constellations  of  Aries  and  Libra,  in  early  ages  they 
defined  the  western  limits  of  those  signs  in  which  the  cor- 
responding constellations  lay,  and  hence  were  designated  as 
the  first  points  of  Aries  and  Libra. 

Owing  to  the  precession  of  the  equinoxes,  the  constellation 
Aries  has  passed  from  the  sign  of  Aries  into  that  of  Taurus, 
but  the  vernal  equinoctial  point,  designated  by  the  sign  T, 
is  still  called  the  "  first  point  of  Aries." 

The  points  of  the  ecliptic  90°  from  the  equinoctial  points 
are  called  solstitial  points,  as  at  these  points  the  sun  reaches 
its  greatest  declination,  occupying  the  northern  one  about 
June  21,  and  the  southern  one  about  December  21;  in  other 
words,  the  obliquity  of  the  ecliptic  equals  the  sun's  greatest 
declination,  North  or  South.  The  hour  circle  passing  through 
the  solstitial  points  is  called  the  solstitial  colure,  PQP'E, 
(Fig.  85).  The  hour  circle  passing  through  the  equinoctial 
points  is  called  the  equinoctial  colure,  POP'  (Fig.  85),  also 
PT  (Figs.  86  and  87).  The  sign  T  stands  for  the  vernal 
equinoctial  point;  the  term  vernal  equinox  refers  to  the  time 
of  the  sun's  passing  through  that  point,  but,  as  custom  sanc- 
tions its  use  to  represent  the  point,  the  term  "  vernal 
equinox"  will  in  future  be  applied  to  the  point,  and  its 
symbol  will  be  T . 


292 


NAUTICAL  ASTSONOMY 


139.  Determination  of  a  point  of  the  celestial  sphere. — The 

position  of  any  point  on  the  surface  of  the  celestial  sphere  is 
determined  when  its  angular  distances  are  given  from  any 
two  great  circles  on  that  sphere,  whose  positions  are  known. 

The  equinoctial  and  ecliptic  are  fixed  great  circles  on  the 
celestial  concave,  and  the  vernal  equinox  is  practically  a  fixed 
point  on  the  equinoctial,  having  a  motion  of  only  50 ''2  a  year 
to  the  westward  due  to  precession.  Each  of  these  great  circles 
is  used  as  the  primary  of  a  system  of  coordinates  in  fixed 
observatories;  but  at  sea  altitudes  are  measured  above  the 
visible  horizon,  and  then  referred  to  the  celestial  horizon,  so 
that  for  seagoing  people  a  system  in  which  the  horizon  is  the 
primary  becomes  necessary.  Hence  three  systems  are  in  use, 
each  named  after  its  primary,  (1)  Ecliptic,  (2)  Equinoctial, 
(3)  Horizon  Systems. 

The  Ecliptic  System  and  Correlated  Terms. 

140.  The  ecliptic  system. — The  primary  circle  of  this  sys- 
tem is  the  ecliptic  which  has  already  been  defined;  the  sec- 
ondaries are  great  circles  passing 
through  the  poles  of  the  ecliptic 
called  circles  of  latitude,  the  one 
passing   through    T,   the  vernal 
equinox,  being  the  principal  one, 
#T  (Fig.  88). 

Celestial  longitude  is  the  arc  of 
the  ecliptic  intercepted  between 
the  vernal  equinox  and  the  circle 
of  latitude  passing  through  the 
body,  reckoned  positively  towards 
the  East,  from  0°  to  360°. 

The  celestial  latitude  of  a  body  is  the  angular  distance  from 
the  plane  of  the  ecliptic  measured  on  a  circle  of  latitude  pass- 
ing through  the  body.  In  Fig.  88,  CO'  is  the  ecliptic; 


THE  EQUINOCTIAL  SYSTEM  293 

the  celestial  longitude;  and  KS,  the  celestial  latitude  of  the 
heavenly  body  8. 

The  coordinates  in  this  system  are  unaffected  by  diurnal 
rotation ;  hence  it  is  a  convenient  system  at  fixed  observatories, 
especially  when  considering  the  motions  of  the  sun  and  bodies 
composing  the  solar  system.  It  is  not  used  at  sea. 

The  Equinoctial  System  and  Correlated  Terms. 

141.  The  equinoctial  system. — In  this  system,  the  primary 
is  the  equinoctial  which  has  already  been  defined,  and  the  sec- 
ondaries are  the  great  circles  passing  through  the  poles  of  the 
equinoctial.  The  solstitial  colure  is  a  secondary  common  to 
this  and  the  ecliptic  system.  The  secondary  of  this  system 
passing  through  the  zenith  of  a  place  is  called  the  celestial 
meridian,  and  that  one  passing  through  a  heavenly  body  is 
called  a  declination  circle. 

The  declination  of  a  heavenly  body  is  its  angular  distance 
from  the  plane  of  the  equinoctial,  measured  on  the  declination 
circle  passing  through  the  body.  It  is  given  in  degrees,  min- 
utes, and  seconds,  and  is  marked  N".  or  S.,  according  as  the 
body  is  North  or  South  of  the  equinoctial  (BA,  Fig.  86). 

The  polar  distance  of  a  heavenly  body  is  its  angular  distance 
from  the  pole  (usually  from  the  elevated  pole),  and,  being 
measured  on  a  declination  circle,  it  equals  90°  —  the  dec- 
lination; but,  if  the  declination  is  negative  (of  an  opposite 
name  from  the  latitude),  the  polar  distance  equals  90°  +  the 
declination. 

Parallels  of  declination  are  small  circles  whose  planes  are 
parallel  to  that  of  the  equinoctial. 

The  rotation  of  the  earth  is  always  performed  in  the  same 
interval  of  time,  a  sidereal  day,  which  is  -divided  into  24  side- 
real hours,  and  gives  to  the  fixed  stars  an  apparent  movement 
in  planes  parallel  to  the  equinoctial,  through  360°  in  the 
same  interval  of  time.  From  the  time  of  apparent  rising  in 


294: 


NAUTICAL  ASTRONOMY 


the  East  till  the  time  of  apparent  setting  in  the  West,  the  stars 
maintain  their  relative  positions  with  reference  to  each  other. 
This  apparent  motion,  being  due  to  the  daily  rotation  of  the 
earth,  is  called  apparent  diurnal  motion  of  the  heavens,  and 
the  path  of  any  one  star  during  its  complete  revolution  is 
called  its  diurnal  circle. 

Right  sphere. — To  an  observer  at  the  equator,  stars  will  rise 
and  set  vertically  and  their  diurnal  circles  will  be  bisected  by 
the  horizon,  so  that  the  stars  will  be  12  hours  above  and  12 


FIG.  89. 


hours  below  the  horizon;  the  planes  of  the  diurnal  circles 
being  at  right  angles  to  the  observer's  horizon,  the  celestial 
sphere  in  this  case  is  called  a  right  sphere. 

Parallel  sphere. — Could  an  observer  be  at  the  North  pole, 
he  would  see  the  stars  of  North  declination  sailing  around, 
maintaining  a  constant  altitude  above  the  horizon,  never  ris- 
ing and  never  setting.  Stars  of  South  declination  would  be 
invisible.  The  planes  of  the  diurnal  circles  being  parallel  to 
the  horizon,  the  celestial  sphere  would  in  this  case  be  called  a 
parallel  sphere. 


THE  OBLIQUE  SPHERE  295 

Oblique  sphere. — To  an  observer  at  some  point  between  the 
equator  and  the  pole,  say  the  North  pole,  the  stars  will 
rise  and  set  at  an  oblique  angle  with  the  horizon.  This 
applies  to  any  heavenly  body,  whose  declination  will  permit 
of  any  part  of  its  diurnal  circle  coming  above  the  horizon. 
A  body  of  0°  declination  will  rise  in  the  East  point,  set  in 
the  West  point,  and  be  the  same  length  of  time  above  and 
below  the  horizon;  EqW  (Fig.  89)  is  the  diurnal  circle  of  such 
a  body. 

A  body  of  North  declination  will  rise  and  set  to  northward 
of  the  East  and  West  points,  and  be  above  the  horizon  more 
than  12  hours.  In  North  latitude,  stars  of  South  declination, 
if  visible  at  a  place  in  North  latitude,  rise  and  set  to  south- 
ward of  the  East  and  West  points,  and  will  be  above  the  hori- 
zon less  than  12  hours.  Since  the  declination  of  the  sun  in 
summer  time  is  of  the  same  name  as  the  elevated  pole,  the  sun 
is  then  above  the  horizon  more  than  12  hours;  in  other  words, 
summer  days  are  longer  than  winter  days.  Those  stars  whose 
polar  distance  is  less  than  the  altitude  of  the  elevated  pole, 
which  is  the  radius  of  the  circle  of  perpetual  apparition,  NK 
(Fig.  89),  never  set,  but  revolve  around  the  elevated  pole  of 
the  heavens.  Those  whose  diurnal  circles  lie  within  the  circle 
of  perpetual  occultation,  RS  (Fig.  89),  never  rise,  and  hence 
are  invisible.  This  aspect  of  the  heavens  is  known  as  the 
oblique  sphere. 

Hour  circles. — In  this  apparent  revolution  of  the  heavenly 
bodies  around  the  earth,  their  declination'  circles  are  continu- 
ously describing  angles  around  the  poles,  which  are  called 
from  the  divisions  of  time  hour  angles,  and,  analogously,  the 
declination  circles  are  called  hour  circles;  hence  hour  circles 
are  defined  as  great  circles  passing  through  the  poles  of  the 
heavens.  PB  (Fig.  86)  is  the  hour  circle  of  the  body  A. 

As  a  star,  for  example  A  (Fig.  86),  moves  in  its  diurnal 
path  about  the  pole,  a  point  B  of  its  hour  circle  moves  uni- 


296  NAUTICAL  ASTRONOMY 

formly  over  the  equinoctial  through  360°  of  arc  in  24  sidereal 
hours,  15°  of  arc  in  one  hour,  15'  of  arc  in  one  minute,  and 
15"  of  arc  in  one  second  of  time,  thus  establishing  a  relation 
between  arc  and  time. 

What  is  said  here  about  the  apparent  movement  of  a  star's 
hour  circle  will  apply  to  the  movements  of  the  hour  circle  of 
any  heavenly  body  whose  increase  of  right  ascension  is  uni- 
form ;  and,  as  time  in  any  system  used  is  the  angle  at  the  pole, 
measured  by  an  arc  of  the  equinoctial,  all  time,  however  meas- 
ured, is  converted  into  arc  at  the  rate  of  15°  of  arc  to  one 
hour  of  time.  See  Art.  178. 

Transit  or  culmination. — The  passage  of  a  celestial  body 
across  the  meridian  of  a  place  is  called  its  transit  or  culmina- 
tion ;  the  upper  transit  occurs  when  it  crosses  the  upper  branch 
of  the  meridian,  and  the  lower  transit  when  it  crosses  the 
lower  branch  of  the  meridian.  When  a  body's  diurnal  path 
is  within  the  circle  of  perpetual  apparition,  both  transits  occur 
above  the  horizon,  the  upper  one  above  the  pole,  the  lower  one 
below  it;  whilst  those  bodies,  whose  diurnal  circles  lie  within 
the  circle  of  perpetual  oceultation,  are  never  visible  at  the 
given  place. 

Hour  angle. — The  hour  angle  of  a  heavenly  body,  or  of  any 
point  of  the  sphere,  is  the  inclination  of  the  hour  (or  declina- 
tion) circle  passing  through  the  body,  or  point,  to  the  celestial 
meridian,  and  is  measured  by  the  arc  of  the  equinoctial  inter- 
cepted between  these  two  circles.  Hour  angles  are  properly 
reckoned  from  the  upper  branch  of  the  meridian,  positively 
toward  the  West,  and  are  usually  expressed  in  hours,  minutes, 
and  seconds  of  time  from  Oh  to  24h.  However,  for  conven- 
ience in  practical  work,  it  is  better,  in  fact  it  is  usual  in  the 
American  naval  service,  to  regard  the  hour  angle  as  minus 
when  the  body  observed  is  East  of  the  meridian  up  to  12h. 
In  Fig.  86,  ZPA  is  the  hour  angle  of  the  body  A  and  it  is 
measured  by  the  arc  CB. 


SOLAR  AND  SIDEREAL  TIME  297 

Solar  time. — The  hour  angle  of  the  sun  is  called  solar  time, 
there  being  24  hours  of  solar  time  in  the  interval  between  two 
consecutive  upper  transits  of  the  sun  over  the  same  meridian, 
and  this  interval  is  called  a  solar  day. 

Sidereal  time. — The  hour  angle  of  the  1st  point  of  Aries,  or 
vernal  equinoctial  point,  is  called  sidereal  time,  there  being  24 
hours  of  sidereal  time  in  the  interval  between  two  consecutive 
upper  transits  of  the  1st  point  o>f  Aries  over  the  same  meridian, 
and  this  interval  is  called  a  sidereal  day.  Owing  to  the  fact 
that  the  1st  point  of  Aries  is  practically  a  fixed  point  of  the 
equinoctial,  the  sidereal  day  is  the  time  of  revolution  of  the 
earth  on  its  axis,  or,  in  other  words,  of  the  apparent  revolution 
of  the  celestial  sphere  through  360°. 

Relation  between  solar  and  sidereal  days. — Owing  to  the 
angular  movement  of  the  sun  in  its  apparent  orbit  to  the  east- 
ward (this  apparent  motion  of  the  sun  being  due  to  the  move- 
ment of  the  earth  in  its  orbit  about  the  sun),  the  sun  comes  to 
the  meridian  each  day  on  an  average  about  3m  5 6s. 555  of  sid- 
ereal time  later  than  on  the  previous  day ;  therefore,  the  solar 
day  is  longer  by  that  amount  than  the  sidereal  day. 

Right  ascension. — The  right  ascension  of  a  heavenly  body 
is  the  inclination  of  its  hour  circle  to  that  passing  through  the 
vernal  equinox,  or  the  arc  of  the  equinoctial  intercepted  be- 
tween these  two  hour  circles.  It  is  measured  from  the  vernal 
equinox  positively  to  the  eastward  from  0  hours  to  24  hours. 
For  body  A  (Fig.  86),  ^PB,  measured  by  the  arc  T#,  is  the 
right  ascension. 

The  fixed  stars  are  at  such  immense  distances  as  to  be  un- 
affected by  the  earth's  change  of  position  in  its  orbit;  the  co- 
ordinates of  this  system,  however,  declination  as  well  as  right 
ascension,  are  slightly  affected  by  the  precession  of  the  equi- 
noxes. 

Relation  of  H.  A.  and  R.  A. — From  the  preceding  defini- 
tions of  hour  angle  and  right  ascension  it  is  evident  from  Fig. 


298  NAUTICAL  ASTRONOMY 

86,  in  which.  A  is  a  heavenly  body,  T  the  1st  point  of  Aries 
or  vernal  equinox,  PZNa  the  meridian,  PB  and  PT  hour 
circles,  that  the  local  sidereal  time  which  equals  the  right 
ascension  of  the  meridian  is  the  angle  ZPT,  measured  by 
T  C;  the  hour  angle  of  the  heavenly  body  A  is  the  angle  ZPA, 
measured  by  CB;  its  right  ascension  is  °rPB,  measured  by 
the  arc  TP,  and  T<7  =  CB  +  °TBf  or  (1)  the  local  sidereal 
time  at  a  given  instant  always  equals  th&  algebraic  sum  of  the 
hour  angle  and  the  right  ascension  of  the  same  ~body  at  that 
instant. 

When  the  hour  angle  is  zero>,  the  heavenly  body  is  on  the 
meridian,  and  its  right  ascension  then  equals  the  local  sidereal 
time  at  that  instant,  or  (2)  the  right  ascension  of  the  meridian 
at  a  given  instant  equals  the  local  sidereal  time. 

These  are  two  facts  that  must  be  fully  realized  and  under- 
stood by  every  navigator ;  and  it  follows  from  the  first  propo- 
sition that  when  two  of  the  angles  are  given,  the  third  can  be 
easily  found. 

The  right  ascension  and  declination  of  heavenly  bodies  are 
determined  at  fixed  observatories,  and  tabulated  in  the  Nau- 
tical Almanacs ;  knowing  these,  the  position  of  a  heavenly  body 
is  easily  determined  in  this  system,  the  right  ascension  being 
reckoned  along  the  equinoctial  to  the  eastward  from  the 
vernal  equinox  in  a  manner  similar  to  the  reckoning  of  longi- 
tude from  the  prime  meridian  on  the  terrestrial  sphere;  and 
the  declination  is  reckoned  North  or  South  of  the  equinoctial 
along  the  declination  circle,  as  latitude  is  reckoned  North  or 
South  of  the  terrestrial  equator  along  a  terrestrial  meridian. 
This  system  is  the  most  convenient  one  for  representing  the 
motions  of  the  fixed  stars,  owing  to  the  very  slight  changes  in 
coordinates. 

The  Horizon  System  and  Correlated  Terms. 

142.  The  horizon  system. — The  primary  circle  of  this  sys- 
tem is  the  celestial  horizon ;  the  secondaries  are  great  circles 


THE  HORIZON  SYSTEM 


of  the  celestial  sphere  passing  through  the  zenith  and  nadir; 
their  planes  being  perpendicular  to  the  horizon,   they   are 
called  vertical  circles.     The  principal  secondary  is  the  celestial 
meridian  which  intersects  the 
horizon    in    the    North    and 
South  points,  each  of  which  is 
named  from  the  nearest  pole. 
The  celestial  meridian  is  the 
secondary  common  to  hoth  the 
horizon    and    equinoctial    sys- 
tems; SZNNa  (Fig.  90). 

The  prime  vertical  is  the 
vertical  circle  pasing  through 
the  E.  and  W.  points  of  the 
horizon ;  its  plane  is,  therefore, 
perpendicular  to  that  of  the  celestial  meridian.  ZWNaE 
(Fig.  90),  is  the  prime  vertical. 

The  azimuth  of  a  heavenly  body  is  the  angle  at  the  zenith, 
measured  by  the  arc  of  the  celestial  horizon,  between  the  me- 
ridian and  the  vertical  circle  passing  through  the  body ;  PZK, 
(Fig.  90),  for  body  A. 

Though  the  azimuth,  as  an  angle  of  the  astronomical  tri- 
angle, is  reckoned  from  the  elevated  pole  towards  the  East  or 
West,  according  as  the  body  is  East  or  West  of  the  meridian, 
and  though  so  estimated  when  tabulated  in  azimuth  tables, 
still  navigators  of  the  present  day  reckon  azimuth  in  'both 
hemispheres  more  conveniently  from  the  North  point  of  the 
horizon,  around  to  the  right,  from  0°  to  360°. 

If  the  angle  Z  found  by  solution  is  x°,  navigators  will  con- 
sider the  azimuth,  or  ZN,  simply  as  x°,  180°  -  —  x°,  180°  +  x° 
or  360°  —  x°,  according  as  the  bearing  of  the  body  by  solution 
is  K  x°  E.,  S.  x°  E.,  S.  x°  W.  or  N  x°  W.,  respectively. 

The  amplitude  of  a  heavenly  body  is  the  angular  distance  of 
the  body,  when  in  the  horizon,  from  the  prime  vertical.  It  is 


300  NAUTICAL  ASTRONOMY 


reckoned  from  the  East  point  when  the  hody  is  rising,  and 
from  the  West  point  when  setting;  towards  North  or  South 
according  as  the  body  is  North  or  South  of  the  prime  vertical. 

The  true  altitude  of  a  heavenly  hody  is  its  angular  distance 
from  the  plane  of  the  celestial  horizon,  measured  on  a  vertical 
circle  passing  through  the  body  from  0°  to  90° ;  KA  (Fig.  90), 
for  body  A. 

The  zenith  distance  is  the  angular  distance  of  the  body  from 
the  zenith,  measured  on  its  vertical  circle,  and  equals  the  com- 
plement of  the  altitude;  ZA  (Fig.  90),  for  body  A. 

From  what  has  been  said  it  follows  that  in  this  system  the 
position  of  a  body  is  given  by  its  altitude  and  azimuth,  the 
coordinates  determined  by  navigators  at  sea,  so  observed  posi- 
tions are  referred  to  this  system ;  but  as  tabulated  elements  to 
be  used  by  navigators  all  over  the  world  must  be  referred  to  a 
system  unaffected  by  the  position  of  the  observer,  the  value 
of  the  equinoctial  system  becomes  apparent. 

The  predicted  positions  according  to  this  latter  system  are 
found  in  the  American  Ephemeris  and  Nautical  Almanac,  as 
well  as  in  other  publications. 

Referring  to  Fig.  90,  let 

0  be  the  observer,  QQf  intersection  of  planes  of  celes- 

Z  the  zenith,  tial  equator  and  meridian. 

tfa  the  nadir,  A  a  heavenly  body, 

P  the  elevated  or  N.  pole,  ZAK  its  vertical  circle, 

SZNNa  the  celestial  meridian,  AK  its  altitude, 

NESW  the  celestial  horizon,  AZ  its  zenith  distance, 

EZ  WNa  the  prime  vertical,  PZK  its  azimuth, 

N  the  North  point  of  the  horizon,  PO  JVthe  altitude  of  the  elevated 

S  the  South  point  of  the  horizon,  pole, 

PPf  the  axis  of  the  sphere,  Ql  0  Z  the  declination  of  the  zenith. 

To  prove  that  latitude  equals  the  altitude  of  the  elevated 
pole. — The  arc  of  the  meridian  SQ'  (Fig.  90),  intercepted  be- 
tween the  planes  of  the  celestial  equator,  QOQ',  and  celestial 
horizon,  SON,  measures  the  inclination  of  the  planes  of  these 


THE  ASTRONOMICAL  TRIANGLE  301 

two  great  circles  to  each,  other;  this  inclination  is  also  meas- 
ured by  the  arc  ZP  intercepted  between  their  poles,  but 
ZP  =  90°  —  PN  and  SQ'  =  90°  —  Q'Z;  therefore, 

PN  =  Q'Z, 
or,  PON  —  Q'OZ. 

Terrestrial  latitude  has  been,  denned  in  Art.  1  as  the  angular 
distance  of  a  place  measured  on  its  meridian  N".  or  S.  of  the 
equator.  '  As  the  zenith  is  the  projection  of  a  place  and  the 
equinoctial  the  projection  of  the  terrestrial  equator  on  the 
celestial  sphere,  the  latitude  of  a  place  is  the  declination  of 
the  zenith;  therefore, 

Lat.  =  Q'OZ  =  PON, 
or,  latitude  equals  the  altitude  of  the  elevated  pole. 

143.  The  astronomical  triangle. — The  spherical  triangle 
'PZA  (Fig.  86),  formed  by  arcs  of  the  celestial  meridian, 
and  the  vertical  and  hour  circles  passing  through  the  body  A, 
is  called  the  astronomical  triangle,  and  it  is  this  triangle  that 
the  navigator  solves  in  working  for  latitude  or  longitude,  re- 
membering that  when  the  observed  body  is  on  the  meridian 
the  triangle  reduces  to  a  straight  line.  The  angles  are :  ZPA 
the  hour  angle,  PZA  the  azimuth,  and  PAZ  the  position 
angle ;  the  sides  of  the  triangle  are  PZ,  the  co-latitude  of  the 
place  of  observation,  AP  the  polar  distance,  and  AZ  the 
zenith  distance  of  the  body.  The  position  angle  is  not  used, 
but  when  any  three  of  the  other  five  parts  are  given,  the  re- 
maining two  can  be  found  by  spherical  trigonometry. 

By  definition  the  co-latitude  and  zenith  distance  can  never 
be  greater  than  90°.  If  the  declination  is  of  the  same  name 
as  the  latitude,  it  is  regarded  as  positive  and  the  polar  distance 
equals  90°  minus  the  declination;  if  of  a  different  name  from 
the  latitude,  the  declination  is  regarded  as  minus  and  the 
polar  distance  equals  90°  plus  the  declination. 

In  studying  the  astronomical  triangle  diagrams  will  be 


302 


NAUTICAL  ASTRONOMY 


found  most  useful,  and  the  most  appropriate  are  those  found 
by  stereographic  projections  in  which  the  point  of  sight  is  at 
one  pole  of  the  primitive  circle.  In  Figs.  91,  92,  and  93, 
PZM  is  a  projection  of  the  astronomical  triangle. 


FIG.  91. 


FIG.  92. 


On  the  plane  of  the  meridian,  the  point  of  sight  is  at  the 
E.  or  W.  point,  and  both  Z  and  P  are  on  the  primitive  circle 
(Fig.  91),  in  which  M  is  a  heavenly  body  West  of  the  meridian. 

On  the  plane  of  the  equator, 
the  point  of  sight  is  at  the  de- 
pressed pole.  P  is  at  the  center 
of  the  primitive  circle,  me- 
ridian, and  declination  or  hour 
circles  are  projected  as  straight 
lines  (Fig.  92). 

On  the  plane  of  the  horizon, 
the  point  of  sight  is  at  the 
nadir.  Z  is  at  the  center  of  the 
primitive  circle.  All  vertical 
circles,  and  hence  the  celestial 
meridian,  are  projected  as 
FIG.  93.  straight  lines  (Fig.  93). 


CHAPTEE  IX. 

THE  SEXTANT,  THE  VERNIER,  AND  THE  ARTIFICIAL 
HORIZON.— METHODS  OF  OBSERVING  HEAVENLY 
BODIES. 

144.  The  sextant. — The  sextant  is  a  small  portable  instru- 
ment used  for  measuring  the  angles  between  two  bodies  or 
objects,  whether  or  not  one  or  both  are  celestial  or  terrestrial, 
and  for  measuring  the  altitudes  of  heavenly  bodies  or  terres- 
trial objects  above  the  visible  horizon.  Its  principal  use  is  at 
sea,  where  the  use  of  fixed  instruments  would  be  impossible, 
in  measuring  altitudes  for  finding  the  latitude  and  longitude. 
The  octant  is  a  similar  instrument,  and  is  used  for  the  same 
purposes,  but  the  length  of  its  limb  is  only  about  one-eighth 
of  a  circle. 

As  the  name  implies,  the  arc  or  limb  (c)  of  the  sextan i 
(Fig.  94)  is  equal  to  about  one-sixth  of  a  circle,  or  60°  of  arc, 
though  graduated,  as  will  be  explained  later  on,  so  that  eacli 
degree  of  the  limb  is  really  divided  into  two  degrees  of  gradu- 
ation, the  subdivisions  of  the  degrees  being  frequently  as  close 
as  10'  of  arc,  on  an  arc  of  silver,  gold,  or  platinum.  The  limb 
and  its  supporting  frame  are  of  brass.  A  brass  index  arm 
(o),  pivoted  at  the  center  of  the  circle  whose  arc  forms  the 
limb,  is  movable,  carrying  at  the  movable  end  a  vernier  (d) 
and  magnifying  glass  (g)  to  read  subdivisions  of  the  gradu- 
ated arc,  and  at  the  pivoted  end  a  silvered  mirror  (a)  whose 
plane  must  be  perpendicular  to  that  of  the  index  arm.  and 
frame.  This  mirror,  called  the  index  glass,  moves  with  the 


304 


NAUTICAL  ASTRONOMY 


index  arm.  A  second  glass  (&),  called  the  horizon  glass,  one- 
half  transparent  and  one-half  silvered,  the  dividing  line  being 
parallel  to  the  plane  of  the  instrument,  is  fixed  and  should  also 
be  perpendicular  to  the  plane  of  the  limb.  The  graduations 


of  limb  and  vernier  should  be  such  that  the  zero  of  one  will 
be  in  coincidence  with  the  zero  of  the  other  when  the  index 
and  horizon  glasses  are  parallel. 
A  telescope  (i)  which  directs  the  line  of  sight  through  the 


THE  SEXTANT 


305 


horizon  glass  and  parallel  to  the  plane  of  the  instrument,  is 
carried  in  a  ring  capable  of  movements  at  right  angles  to-  the 
plane  of  the  instrument,,  shifting  the  axis  of  telescope  from 
the  silvered  to  the  transparent  part  of  the  horizon  glass,  or  vice 
versa. 

Colored  glasses  (h)  of  different  shades  are  fitted  for  use 
before  both  index  and  horizon  glasses. 

The  index  arm  is  fitted  with  a  clamp  (e)  for  securing  it  to 


FIG.  95. 


the  limb,  and  a  tangent  screw  (/)  for  giving  it  small  motions 
after  clamping. 

Besides  the  telescope  (i),  the  sextant  box  is  usually  fitted 
out  with  a  star  or  inverting  telescope  (k),  a  plain  or  sighting 
tube  (Z),  and  neutral  glasses  or  caps  (n)  for  the  telescopes. 
The  use  of  these  caps  obviates  the  necessity  for  the  use  of  the 
colored  shade  glasses.  The  box  also  contains  a  screw  driver, 
adjusting  keys,  a  magnifying  glass,  and  spare  mirrors. 


306  NAUTICAL  ASTRONOMY 

145.  The  optical  principle  of  the  sextant,  —  The  optical 
principle  of  the  construction  of  the  sextant  is  thus  stated: 
"  The  angle  between  the  first  and  last  directions  of  a  ray  of 
light,  which  has  suffered  two  reflections  in  the  same  plane,  is 
equal  to  twice  the  angle  which  the  two  reflecting  surfaces 
make  with  each  other  " 

To  prove  this,  let  M  and  m  be  the  two  reflecting  mirrors  of 
a  sextant  whose  planes  are  perpendicular  to  the  plane  of  the 
sextant,  in  this  case  the  plane  of  the  paper  (Fig.  95).  Let  B 
be  a  body  whose  ray  falling  on  M  is  reflected  to  m  and  by  m  to 
the  eye  at  E;  then  BEm  will  be  the  angle  between  the  first  and 
last  directions  of  ray  BM,  after  having  been  reflected  twice  in 
the  same  plane.  The  angle  between  the  mirrors  is  equal  to  the 
angle  between  lines  perpendicular  to  them,  ppf  being  perpen- 
dicular to  M  and  mpf  to  m,  and  it  is  required  to  prove  that 
BEm  or  h  =  2a.  Since  the  angle  of  incidence  equals  the 
angle  of  reflection,  BMp  •=  pMm,  and  Mmp'  =  p'mE  ; 
by  geometry 

from  kMp'm,  x  =  y  -(-  a  .  '  .  2x  =  %y  -\-  2a 
from  &MEm,  2x  =  2y  +  h 


therefore  h  =  2a 

146.  Application  of  the  principle  in  measuring  angles.  — 

Suppose  it  is  desired  to  measure  the  angular  distance  between 
two  bodies,  B  and  H,  H  sufficiently  distant  that  the  rays  H'M 
and  Hm  are  sensibly  parallel.  The  instrument  is  held  so  that 
its  plane  passes  through  both  objects,  the  object  H  being  seen 
directly  through  the  telescope  and  horizon  glass.  Now  let  the 
index  arm  be  so  placed  and  clamped  that  the  two  glasses  are 
parallel  to  each  other;  then  will  the  ray  H'M  be  reflected  by 
the  two  glasses  parallel  to  itself,  and  the  observer's  eye  at  E 
will  see  both  direct  and  reflected  images  in  coincidence.  Sup- 
pose this  position  of  the  index  arm  is  MI,  then  for  the  given 
position  of  the  horizon  glass,  /  should  be  the  zero  of  gradua- 
tions of  the  limb.  Now  move  the  index  bar,  and  with  it  the 


THE  VERNIER  307 

fixed  mirror  M,  to  the  position  MI',  so  that  a  ray  from  the 
second  object  B  shall  be  reflected  in  the  direction  mE;  the 
observer  looking  directly  at  H  through  the  transparent  part  of 
the  horizon  glass,  sees  the  reflected  image  of  B  in  coincidence 
with  the  direct  image  of  H.  The  angle  h  is  the  angle  meas- 
ured, but  h  is  twice  the  angle  between  the  mirrors,  or  Ji  =  2a ; 
and,  since  a  =  IMI'9  h  is  twice  the  angle  through  which  the 
index  bar  has  moved,  that  is,  twice  the  difference  of  the  read- 
ings I  and  /'.  To  avoid  doubling  the  angle,  every  half  degree 
of  II',  and  in  fact  of  the  whole  limb,  is  marked  as  a  whole 
degree,  and  the  observer,  reading  directly  from  the  limb,  has 
only  to  subtract  the  reading  at  I  from  that  at  I',  to  get  the 
angular  distance  between  H  and  B.  If  the  instrument  is  in 
proper  adjustment,  the  reading  at  /  is  zero,  that  is,  the  limb 
is  graduated  from  I  as  an  origin.  If  this  point  of  reference, 
I,  does  not  coincide  with  the  zero  of  graduation,  the  sextant 
has  an  error,  called  index  error,  which  affects  all  angles  ob- 
served with  it  at  the  time. 

The  degrees  of  the  limb  are  further  subdivided,  those  of 
the  finest  sextants  being  divided  inttf  six  equal  parts,  each  part 
10'  of  arc,  and  in  order  to  read  fractions  of  these  divisions, 
recourse  is  had  to  the  vernier. 

147.  The  vernier. — This  is  a  graduated  scale  (Fig.  96)  to 
slide  along  the  divisions  of  a  graduated  limb  to  facilitate  the 
readings  to  fractions  of  a  division  of  the  limb.  It  is  so  con- 
structed that  the  length  of  the  vernier  is  exactly  the  length  of 
a  certain  integral  number  of  divisions  of  the  limb,  and  is 
divided  into  one  more  or  less  divisions  than  that  certain  num- 
ber ;  the  fraction  of  a  division  of  the  limb  is  indicated  by  the 
division  of  the  vernier  which  is  in  coincidence  with  a  division 
of  the  limb,  as  will  be  explained  later.  The  most  usual  method 
of  construction  is  to  make  the  number  of  divisions  on  the 
vernier  one  more  than  on  the  corresponding  arc  of  the  limb, 
and  the  explanation  of  this  type  follows. 


308 


NAUTICAL  ASTEONOMY 


To  explain  the  working  of  a  vernier,  let  AB  (Fig.  96)  be 
the  arc  of  a  limb,  each  division  20' ;  CD  the  vernier,  the  length 
of  which  is  taken  as  19  times  the  length  of  a  division  of  the 
limb  and  is  divided  into  20  equal  parts,  thus  each  division  of 
the  vernier  comprises  19'  of  arc  or  is  less  by  1'  of  arc  than  any 
division  of  the  limb.  The  first  line  of  the  vernier  is  the  zero 
line,  and  the  reading  of  the  limb  is  determined  by  the  posi- 
tion of  this  zero.  If  this  zero  coincides  with  any  division  of 
the  limb,  the  division  line  of  the  vernier  marked  1  falls  short 
of  the  next  division  of  the  limb  by  1',  the  next  division  line  of 
the  vernier  marked  2  falls  short  of  the  next  line  of  limb  by  2', 
and  so  on  until  the  line  marked  20  of  the  vernier  coincides 
with  a  line  of  the  limb;  hence,  if  the  vernier  is  advanced 


FIG.  96.      . 

through  1'  of  arc,  the  line  marked  1  of  the  vernier  will  coincide 
with  a  division  of  the  limb,  if  we  advance  it  through  2'  of  arc, 
the  line  marked  2  will  coincide  with  a  division  of  limb,  and 
so  on,  and  if  the  nth  line  of  the  vernier  is  found  to  be  in 
coincidence  with  a  division  of  the  limb,  it  will  be  evident 
that  the  zero  of  the  vernier  has  advanced  n  minutes. 

General  rule  for  navy  sextants. — The  general  rule  fol- 
lowed in  the  construction  of  verniers  for  the  U.  S.  Navy  is 
to  take  the  length  of  the  vernier  exactly  equal  to  the  length 
of  a  certain  integral  number  of  divisions  of  the  limb  and  di- 
vide, the  vernier  length  into  equal  parts,  the  number  of  which 


HEADING  THE  SEXTANT  309 

must  be  greater  by  one  than  the  number  of  the  divisions  of 
the  limb. 

Let  I  =  value  of  a  division  of  the  limb, 
v  =  value  of  a  division  of -the  vernier, 
n  =  number  of  parts  into  which  the  vernier  is  divided, 
n  —  1  =  number  of  parts  in  the  corresponding  length  of 
the  limb, 


n  n  n> 

Least  count  of  vernier  =  I  - 

Of  course,  the  graduations  of  the  limb  and  the  vernier  must 
be  in  the  same  unit. 

Ex.  70. — The  limb  of  a  sextant  is  divided  to  10'  of*  arc. 
Construct  a  vernier  to  read  to  10"  of  arc. 

The  least  count  being  10",  a  division  of  the  limb  10'  =  600", 
a  division  of  the  vernier  is  590".  Therefore, 

600  (n—1)  =  590  nf 
60  n  —  59  n  =  60,  n  =  60. 

Take  59  divisions  of  the  limb  for  the  length  of  the  vernier, 
and  divide  it  into  60  equal  parts. 

Ex.  11. — A  sextant  limb  reads  to  15'  of  arc;  the  vernier  is 
taken  in  length  as  44  divisions  of  the  limb.  What  is  the  least 
count  of  the  vernier  ? 


148.  Reading  the  sextant. — First  note  the  position  of  the 
zero  of  the  vernier,  then  read  the  limb  up  to  the  division  line 
immediately  to  the  right  of  the  zero  of  the  vernier;  this  will 
be  a  certain  number  of  degrees,  or  a  certain  number  of  degrees 
plus  a  certain  number  of  the  divisions  of  a  degree.  Say  the 
sextant  limb  is  graduated  to  10'  of  arc,  and  suppose  the  near- 
est division  referred  to  is  33°  40';  now  follow  the  arc  of  the 
vernier  until  a  vernier  line  is  found  in  coincidence  with  any 


310  NAUTICAL  ASTRONOMY 

line  of  the  limb.  Suppose  the  least  count  of  the  vernier  is 
10"  of  arc,  and  the  reading  of  the  vernier  at  the  coincident 
line  is  2'  20",  then  the  angle  is  33°  42'  20".  All  angles 
measured  on  the  limb  are  spoken  of  as  "  on  the  arc." 

Excess  of  arc. — The  limb  of  a  sextant  is  generally  grad- 
uated not  only  to  120°,  but  the  limb  is  often  of  such  extent 
as  to  be  graduated  up  to  150°.  This  part  of  the  limb  is  the 
arc  proper,  but,  in  all  sextants,  the  limb  and  graduated  arc  are 
continued  to  the  right  beyond  the  zero  for  a  short  distance, 
and  this  arc  is  called  the  "excess  of  arc,"  and  angles  meas- 
ured on  it  are  spoken  of  as  "  off  the  arc."  These  angles  are 
read  from  zero  to  the  right,  or  backwards,  and  the  vernier 
must  also  be  read  backwards.  If  a  division  of  the  limb  is 
n  minutes,  then  the  vernier  is  marked  to  read  n  minutes;  so 
read  the  vernier  directly  and  subtract  its  reading  from  n  min- 
utes to  get  the  vernier  reading  to  be  added  to  the  reading  of 
the  limb,  off  the  arc,  immediately  to  the  left  of  the  zero  of  the 
vernier.  Thus  if  a  sextant  limb  reads  to  10',  the  vernier  to 
10",  and  the  zero  of  the  vernier  falls  between  3°  10'  and  3°  20' 
off  the  arc,  and  the  vernier,  if  read  directly,  shows  2'  50",  then 
the  vernier  read  backwards  would  be  7'  10",  and  the  angle  3° 
17'  10". 

149.  Errors. — Sextants  are  subject  to  two  general  classes 
of  errors.  The  first  class  comprises  what  are  known  as  con- 
stant errors,  and,  though  they  usually  arise  from  defects  in 
construction,  they  may  at  times  be  occasioned  by  injuries  re- 
ceived in  legitimate  use,  or  to  abuses  due  to  ignorance  or 
carelessness.  These  errors  should  be  eliminated,  or  ascer- 
tained and  tabulated  by  the  maker.  In  a  high-grade  instru- 
ment from  a  maker  of  good  reputation,  this  class  of  errors 
should  not  exist. 

The  second  class,  known  as  the  adjustment  class,  comprises 
those  errors  that  should  be  removed  by  the  navigator  himself. 

Constant  errors. — (1)  The  centering  error  is  due  to  the 


ERRORS  or  SEXTANT  311 

fact  that  the  axis  of  the  index  bar  is  not  at  the  center  of  the 
limb  nor  perpendicular  to  its  plane.  ~No  sextant  should  be 
bought  without  careful  inspection  and  not  until  after  tests  as 
to  the  centering  error  have  been  made.*  If  the  eccentricity  is 
found  to  be  greater  than  .005  of  an  inch,  the  instrument  should 
be  rejected.  (2)  Error  of  graduation.  The  limb  may  not  be 
a  plane  surface,  and  graduations  of  both  limb  and  vernier  may 
be  inex-act.  There  may  be  flexure  of  limb  due  to  varying  tem- 
perature, or  accidental  blows,  producing  great  errors  in  angles. 
(3)  Prismatic  effect  of  mirrors  and  shade  glasses,  due  to  a 
want  of  parallelism  between  the  two  faces. 

The  above  are  all  faults  of  construction. 

The  combined  total  errors  of  eccentricity  and  graduation 
can  be  ascertained  together  by  measuring  known  angles  with 
the  sextant ;  the  error  can  be  found  for  a  number  of  positions 
of  the  index  bar,  and  then  for  other  intermediate  angles  by 
interpolation.  The  known  angles  referred  to  may  consist  of 
angles  laid  off  by  a  theodolite  at  intervals  of  10°  to  20°  to 
cover  the  range  of  the  sextant. 

The  combined  error  can  also  be  ascertained  hy  a  series  of 
artificial  horizon  observations,  observing  stars  of  nearly  equal 
altitudes  N".  and  S.  of  the  zenith.  Half  the  difference  of 
latitudes  resulting  from  each  star  will  be  the  error  for  that 
altitude.  The  correction  will  be  minus,  if  the  latitude  from 
the  star  on  the  polar  side  of  the  observer  is  greater  than  that 
from  the  star  on  the  equatorial  side  of  the  observer ;  and  plus, 
if  vice  versa.  As  this  error  varies  on  different  parts  of  the 
arc,  and  generally  increases  with  the  angle,  it  would  require 
many  observations  to  determine  it  with  any  degree  of  satis- 
faction. 

The  determination  of  this  error  at  sea  is  an  entirely  differ- 
ent proposition ;  theoretically  it  can  be  done  by  measuring  the 
angular  distance  between  two  stars,  and  comparing  this  with 
the  angular  distance  ascertained  by  computation.  This,  how- 

*  Navigating  sextants  issued  to  the  U.  S.  Navy  should  bear  a  certificate  of  in- 
spection from  the  U.  S.  Naval  Observatory,  giving  the  correction,  for  eccentricity 
at  intervals  of  10"  of  arc. 


312  NAUTICAL  ASTRONOMY 

ever,  is  not  practicable  on  account  of  the  complications  due 
to  refraction  and  aberration. 

However,  since  sextants  are  liable  to  accidents  at  sea,  it 
may  be  desirable  to  ascertain  this  error,  if  only  approximately. 
Now,  if  we  can  observe  the  angular  distance  between  stars  on 
the  same  vertical  circle,  the  question  of  refraction  will  become 
a  very  simple  matter,  as  the  altitudes  may  be  either  observed 
(the  horizon  being  clear),  or  computed  for  the  instant  of 
measurement  of  the  arc.  Those  stars  that  have  practically 
the  same  right  ascension,  or  right  ascensions  differing  by 
12  hours,  will  be  on  the  meridian,  and  hence  on  a  vertical 
circle  at  the  same  time.  There  are  many  groups  of  such  stars 
whose  right  ascensions  do  not  differ  more  than  either  30 
minutes,  or  12h  ±  30m,  and  these  might  be  used  without  much 
error.  However,  the  right  ascensions  of  the  following  groups 
are  practically  the  same 

a  Aurigse      )    77  Ursae  Majoris    1 

ft  Orionis      \    a  Virginis  j 

a  Scorpii      |    a  Pavonis 

s  Ophiuchi  J    y  Cygni 

and  the  right  ascensions  of  the  groups  below  differ  by  prac- 
tically 12  hours, 

a  Ursas  Minoris  j  a  Tauri  j  y  Geminorum  ) 

a  Virginis  J  a  Trianguli  Australis  J  a  Lyrse  J 

and  the  stars  of  each  pair  are  on  the  meridian  at  practically 
the  same  time.  Now  the  true  distance,  within  an  error  of  a 
few  seconds  of  arc  only,  neglecting  aberration,  between  the 
stars  in  each  of  the  first  four  groups,  that  is,  stars  of  the  same 
right  ascension,  is  the  difference  of  their  polar  distances  at  the 
instant  of  meridian  passage;  for  stars  of  the  second  groups, 
that  is,  for  stars  whose  right  ascensions  differ  by  12  hours,  the 
true  distance  will  be  the  sum  of  their  polar  distances. 

If  any  stars,  paired  off  as  above,  are  visible,  measure  the 


PRISMATIC  EFFECTS  313 

arc  between  them,  when  on  the  meridian;  either  observe  or 
compute  their  altitudes,  and  take  from  tables  the  correspond- 
ing refractions.  If  both  stars  are  on  the  same  side  of  the 
zenith,  add  the  difference  of  refractions  to  the  observed  arc; 
if  the  stars  are  on  opposite  sides  of  the  zenith,  add  the  sum 
of  refractions  to  the  observed  arc ;  the  result  is  the  corrected 
sextant  distance;  the  difference  between  this  and  the  true 
distance  obtained  from  the  polar  distances  is  the  total  error 
for  that  angle.  Knowing  the  index  error,  the  error  due  to 
eccentricity  and  graduation  may  be  found. 

Knowing  the  right  ascension  of  a  star,  it  is  easy  to  find 
the  ship's  time  of  its  transit,  and  hence  the  time  for  measur- 
ing the  arc.  However,  for  many  apparent  reasons,  even  this 
method  is  ordinarily  impracticable,  so  that  at  sea  the  sextant 
should  be  guarded  carefully  against  all  possible  injuries. 

Graduations. — Examine  carefully  the  graduations  of  both 
limb  and  vernier.  If  the  zero  of  the  vernier  is  in  coincidence 
with  one  division  of  the  limb,  an  inspection  of  the  divisions 
of  the  vernier  should  show  an  increasing  separation  between 
the  divisions  of  vernier  and  limb  in  a  direction  towards  the 
zero  of  vernier  till  the  last  division  of  the  vernier  is  reached, 
when  it  should  be  coincident  with  a  division  of  the  limb. 
By  shifting  the  position  of  the  vernier,  the  divisions  of  the 
limb  are  tested  for  equality,  whilst  for  magnitude  they  may 
be  tested  by  measuring  known  angles  of  various  sizes.  Faults 
of  graduation,  if  developed  on  inspection  before  buying, 
should  cause  the  rejection  of  the  sextant. 

Prismatic  effect  of  index  glass. — This  can  be  tested  by  ob- 
serving a  large  angle,  say,  120°  to  130°,  between  two  objects, 
and  again  measuring  the  angle  with  the  index  glass  upside- 
down.  If  measurements  agree,  the  sextant  having  been  ad- 
justed in  both  cases,  there  is  no  prismatic  error.  If  they  do 
not  agree,  reject  the  mirror.  When  a  reflected  image,  the 
angle  being  large,  is  not  clearly  defined,  or  there  seems  to  be 


314  NAUTICAL  ASTRONOMY 

a  fainter  outline  on  a  clearer  image,  it  is  evident  that  rays 
reflected  from  the  inner  and  outer  faces  of  the  index  glass 
are  not  parallel,  and  that  the  glass  is  prismatic. 

Prismatic  effect  of  horizon  glass. — The  want  of  parallel- 
ism of  the  two  faces  of  the  horizon  glass  is  not  a  matter  of 
great  importance,  as  all  angles  and  the  index  correction  are 
affected  alike. 

Prismatic  effect  of  shade  glasses. — A  want  of  parallelism 
in  shade  glasses,  when  used  in  front  of  the  index  glass,  will 
affect  the  index  correction,  which  should  be  determined  with 
and  without  them.  The  index  error  should  be  determined 
also  for  each  combination  of  shade. 

These  shade  glasses,  when  known  to  be  prismatic,  should 
be  discarded;  and,  if  thought  to  be  prismatic,  colored  caps 
should  be  put  on  the  telescopes  and  the  use  of  shade  glasses 
discontinued. 

Imperfections  of  shade  glasses  between  the  eye  and  horizon 
glass,  or  in  the  colored  cap  of  the  telescope,  affect  the  object 
and  the  reflected  image  alike,  so  that  the  angle  between  them  is 
unaffected. 

150.  Adjustment  of  the  sextant. — It  is  the  duty  of  the 
navigator,  or  of  the  person  using  a  sextant,  to  keep  it  in  ad- 
justment; in  other  words,  to  see  that  the  index  and  horizon 
glasses  are  both  perpendicular  to  the  plane  of  the  instrument 
and  parallel  to  each  other  when  the  zeros  of  the.  vernier  and 
limb  are  in  coincidence,  and  that  the  line  of  sight  of  the  tele- 
scope is  parallel  to  the  plane  of  the  instrument. 

To  adjust  the  index  glass. — Hold  the  sextant  in  the  left 
hand,  place  the  index  bar  near  the  center  of  the  limb,  and, 
with  the  index  glass  nearest  the  eye,  the  eye  being  near  the 
plane  of  the  instrument,  look  into  the  mirror  so  as  to  see  the 
reflected  image  of  the  limb.  If  the  image  and  the  arc  as 
seen  direct  form  a  continuous  line,  the  adjustment  is  correct. 
If  they  form  a  broken  line,  the  mirror  inclines  forward  or 


ADJUSTMENT  OF  SEXTANT  315 

backward,  according  as  the  image  rises  or  droops.  Adjust  the 
glass  by  means  of  screws  at  the  back  of  the  glass  till  the  arc 
and  its  image  appear  perfectly  continuous. 

To  perfect  the  adjustment,  it  may  be  necessary  in  some  cases 
to  tilt  the  mirror  so  much  as  to  require  the  use  of  a  liner  of 
blotting  paper  under  one  edge. 

To  adjust  the  horizon  glass. — The  horizon  glass  may  pro- 
duce two  kinds  of  error,  a  lateral  error  and  an  index  error. 

Place  the  zeros  of  vernier  and  limb  in  coincidence,  and  look 
through  the  telescope  at  a  star.  If  the  two  images  coincide, 
the  adjustment  is  correct.  If  the  reflected  image  is  to  the 
right  or  left  of  the  direct  image,  there  is  lateral  error  due  to 
the  fact  that  the  horizon  glass  is  not  perpendicular  to  the 
frame;  if  the  reflected  image  is  above  or  below  the  direct 
image,  there  is  index  error  due  to  the  fact  that  the  mirrors 
are  not  parallel.  The  adjusting  screws  for  this  glass  are 
sometimes  back  of  the  glass,  at  other  times  below  and  to  one 
side.  Move  the  arm  and  bring  the  mirrors  parallel  so  as  to 
have  the  reflected  image  on  the  same  line  but  to  right  or  left 
of  the  direct  one.  By  proper  screws  remove  the  lateral  error, 
so  that,  as  the  arm  is  moved,  the  reflected  image  passes  directly 
over  the  direct  image.  Now  place  the  zeros  in  coincidence, 
and,  by  the  proper  screws,  make  the  two  images  coincide, 
thereby  eliminating  the  index  error.  However,  these  two 
errors  are  so  intimately  connected  that  in  an  effort  to  remove 
one,  the  other  is  affected;  so  it  is  better  to  adjust  by  the 
"halving  method."  Place  the  zeros  in  coincidence,  remove 
half  the  lateral  error,  then  half  the  index  error,  and  so  on 
till  adjustment  is  perfect. 

The  sea  horizon  may  be  used  to  test  the  adjustment  of  the 
horizon  glass  as  follows:  hold  the  instrument  vertically 
and  make  the  reflected  and  direct  image  of  the  horizon  a  con- 
tinuous line.  Then  incline  the  instrument  so  that  its  plane 
makes  but  a  small  angle  with  the  plane  of  the  horizon.  If 


316  NAUTICAL  ASTRONOMY 

the  true  and  reflected  horizons  are  in  perfect  continuation, 
each  of  the  other,  the  glass  is  perpendicular  to  the  plane  of 
the  instrument,  the  reading  of  limb  and  vernier  being  the 
index  error.  If  the  reflected  horizon  appears  above  the  true 
one,  the  glass  leans  too  much  inward,  otherwise  outward. 

The  sun  may  be  used  in  the  same  way  as  the  star  was  used, 
but,  owing  to  its  size  and  brightness,  perfection  of  adjustment 
is  not  so  easily  reached. 

To  make  the  line  of  sight  of  telescope  parallel  to  the  plane 
of  the  instrument. — Screw  the  inverting  telescope  into  the 
collar,  turn  the  eye  tube  till  the  two  wires  at  its  focus  are 
parallel  to  the  plane  of  the  instrument.  Place  the  sextant 
upon  a  table  in  a  horizontal  position,  look  along  the  plane 
of  the  limb,  and  make  a  mark  upon  a  wall,  or  other  vertical 
surface,  at  a  distance  of  about  20  feet;  draw  another  mark 
above  the  first  at  a  distance  equal  to  the  height  of  the  axis 
of  the  telescope  above  the  plane  of  the  limb;  then  so  adjust 
the  telescope  that  the  upper  mark,  as  viewed  through  the 
telescope,  falls  midway  between  the  wires.  The  adjustment  is 
made  by  tightening  or  loosening  one  of  the  two  adjusting 
screws  on  the  collars,  doing  the  reverse  with  the  other 
screw. 

Index  error. — Before  using  a  sextant  to  make  observations 
of  any  kind,  the  sextant  being  otherwise  in  adjustment,  it  is 
necessary  to  find  the  point  of  the  graduated  arc  where  the 
zero  of  the  vernier  falls  when  the  two  mirrors  are  parallel 
to  each  other,  or  at  the  time  when  the  reflected  image  of  a 
distant  object  is  found  to  be  coincident  with  the  direct  image 
of  the  object  itself.  If  this  point  is  not  coincident  with  the 
zero  of  the  limb,  the  sextant  has  an  index  error  which  affects 
all  angles  taken  at  the  time.  These  angles,  as  measured,  will 
be  too  small  or  too  large,  according  as  the  zero  of  the  vernier 
falls  to  the  right  or  to  the  left  of  the  zero  of  the  arc ;  in  other 
words,  the  reading  of  the  vernier  is  subtractive,  if  on  the  arc, 


INDEX  CORRECTION  317 

additive,  if  off  the  arc.  The  error  applied  in  this  way  is 
known  as  the  index  correction,  and  is  represented  by  the  let- 
ters I.  C. 

Should  it  be  desired  to  eliminate  the  index  error,  place 
the  zeros  of  vernier  and  limb  in  coincidence,  and,  by  means 
of  the  proper  adjustment  screws,  turn  the  horizon  glass 
about  an  axis  perpendicular  to  the  plane  of  the  instrument 
till  the  reflected  and  direct  images  of  a  star  or  distant  object 
are  in  coincidence.  Be  careful,  however,  not  to  disturb  the 
perpendicularity  of  the  horizon  glass.  It  is  not  advisable  to 
try  to  keep  the  index  error  at  zero,  but  it  should  be  deter- 
mined before  every  observation  under  any  circumstances ;  and 
•the  knowledge  that  there  is  one,  even  though  small,  makes  its 
determination  necessary  at  the  time  of  any  set  of  observations. 

To  determine  the  index  correction. — By  a  star. — Bring  the 
reflected  image  of  the  star  into  coincidence  with  its  direct 
image,  then  read  the  arc  and  vernier.  The  reading  is  the 
index  correction ;  -f-  if  off  the  arc,  —  if  on  the  arc. 

By  the  sea  horizon. — Hold  the  instrument  vertical,  and 
make  the  true  and  reflected  horizons  continuous.  The  read- 
ing of  the  arc  and  vernier  is  the  correction,  -|-  or  —  as  before. 

By  the  sun. — Bring  upper  limb  of  reflected  image  of  sun 
tangent  to  the  lower  limb  of  the  sun  seen  directly,  read  the 
sextant,  +.  if  off  the  arc,  —  if  on  the  arc;  then  bring  the 
lower  limb  of  the  reflected  image  tangent  to  the  upper  limb 
of  the  sun  seen  directly,  read  the  sextant,  +  if  off,  —  if  on  the 
arc.  The  index  correction  will  be  one-half  the  algebraic  sum 
of  the  two  readings.  If  well  taken,  the  result  can  be  checked 
by  finding  from  the  Almanac  the  sun's  semidiameter  which 
should  equal  one-fourth  ftie  algebraic  difference  of  the  two 
readings.  However,  since  refraction  acts  in  the  vertical 
plane,  and  affects  the  vertical  diameter  of  the  sun,  the  amount 
of  course  depending  on  the  altitude,  it  is  preferable  at  low 
altitudes  to  use  the  horizontal  diameter  in  finding  the  index 


318  NAUTICAL  ASTRONOMY 

correction;  but,  as  the  difference  of  the  refractions  for  the 
•upper  and  lower  limbs  of  the  sun,  for  altitudes  over  30  °, 
amounts  to  only  a  few  seconds  of  arc,  there  is  no  practical 
advantage  in  using  the  horizontal  diameter  at  the  higher 
altitudes. 

151.  Using  a  sextant.  To  observe  at  sea  an  altitude  of 
the  sun,  in  other  words,  the  angular  distance  along  a 
vertical  circle  from  the  sun  to  the  horizon:  adjust  the 
telescope  to  distinct  vision  by  looking  through  the  tele- 
scope at  the  horizon,  moving  the  eye  piece  in  or  out  till 
the  horizon  is  clearly  and  distinctly  seen,  then  screw  it  into 
its  collar;  see  the  instrument  in  adjustment  and  ascertain 
the  index  correction;  put  down  the  necessary  shade  glasses 
before  the  index  glass,  place  the  index  bar  near  the  zero  of 
the  limb,  and  see  the  tangent  screw  in  mid  position ;  hold  the 
instrument  in  the  right  hand  by  the  handle  so  that  its  plane 
shall  be  vertical,  and  direct  the  line  of  sight  'to  a  point  of  the 
horizon  directly  below  the  sun.  Now  move  the  index  bar 
with  the  left  hand,  and,  if  the  sextant  is  held  properly  when 
its  reading  is  near  the  altitude  of  the  sun,  its  reflected  image 
will  be  seen  to  descend.  Make  an  approximate  contact  with 
the  horizon  and  clamp  the  index  bar.  Now  rotate  the  sex- 
tant slightly  around  the  line  of  sight  as  an  axis,  making  the 
reflected  image  skirt  along  the  horizon;  and,  by  means  of  the 
tangent  screw,  find  one  point  at  which  the  sun  is  just  tangent 
to  the  horizon ;  the  reading  of  the  sextant  at  that  time  is  the 
altitude  of  the  sun's  lower  limb.  Just  before  the  altitude  is 
taken  tell  the  assistant  to  "  stand  by,"  and  at  the  instant  when 
the  altitude  is  taken,  say  "  mark."  *  The  assistant  notes  the 
seconds,  minutes,  and  hours  of  his  watch,  recording  opposite 
the  time  the  degrees,  minutes,  and  seconds  of  the  altitude. 
At  the  time  "mark/'  the  sun  should  be  at  the  lowest  point 
of  its  arc,  just  tangent  to  the  horizon  and  in  the  center  of  the 

*An  experienced  observer  should  be  able  to  note  the  time  of  his  own 
observations. 


OBSERVING  ALTITUDES  319 

field  of  view.  If  by  inclining  the  sextant,  the  sun  is  moved 
from  this  center  to  points  nearer  the  plane  of  the  instrument, 
or  farther  off,  the  angle,  as  read  on  the  arc,  will  be  too  great. 
If  there  is  much  glare  around  the  horizon,  as  is  frequently  the 
case  when  the  sun  is  observed,  especially  at  low  altitudes, 
shade  glasses  should  be  turned  down  in  front  of  the  horizon 
glass  as  may  be  necessary.  The  amount  of  light  and,  hence, 
the  brightness  of  the  reflected  object  can  be  varied  by  moving 
the  telescope  towards  or  from  the  plane  of  the  sextant. 

To  observe  the  altitude  of  a  star. — In  observing  a  star, 
the  observer  can  use  the  inverting  telescope,  which  has  greater 
magnifying  power  than  the  ordinary  direct  one,  or  he  can 
use  the  plain  tube.  Place  the  zero  of  the  vernier  on  the  zero 
of  the  limb,  look  at  the  star,  hold  the  instrument  vertical, 
move  the  index  bar  outward,  keeping  the  reflected  image  in 
sight  till  the  image  of  the  star  is  just  below  the  horizon; 
clamp,  and  whilst  rotating  the  instrument,  use  the  tangent 
screw  and  find  the  lowest  point  of  the  swing  just  on  the  hori- 
zon. It  is  advisable  not  to  use  any  telescope  or  tube  until 
able  to  observe  well  without  it.  In  bringing  down  and  ob- 
serving a  star  with  a  tube  or  the  unassisted  eye,  keep  both 
eyes  open.  This  method  is  essential  to  avoid  bringing  down 
the  wrong  star,  and  it  might  be  used  for  the  sun  with  begin- 
ners, though  it  would  be  very  trying  on  the  eye. 

Sometimes  when  latitude  is  approximately  known,  and  an 
observation  of  a  star  on  the  meridian  is  to  be  made,  the  instru- 
ment can  be  advantageously  set  to  the  approximate  altitude. 

An  observer  can  determine  his  personal  error  in  measuring 
altitudes  of  stars  by  taking  several  altitudes  of  Polaris  for 
latitude  at  a  place  whose  latitude  is  accurately  known. 

To  measure  the  angle  between  two  visible  objects. — Direct 
the  line  of  sight  (or  the  telescope  which  should  be  and  can 
be  easily  used  after  a  little  practice)  toward  the  left  hand 
object,  if  both  are  nearly  in  the  same  horizontal  plane,  or 


320  NAUTICAL  ASTRONOMY 

toward  the  lower  one,  if  one  is  much  above  the  other.  With 
this  object  in  direct  view,  through  the  plain  part  of  the  hori- 
zon glass,  move  the  index  arm  until  the  image  of  the  other 
object,  after  reflection  by  the  index  glass,  is  seen  in  the  sil- 
vered portion  of  the  horizon  glass.  Having  made  a  partial 
contact  of  the  two  images,  clamp  the  instrument,  screw  in  the 
telescope  (if  not  already  in  its  collar),  perfect  the  contact 
with  the  tangent  screw,  and  then  read  the  limb. 

If,  for  any  reason,  it  should  be  desirable  to  point  the  tele- 
scope to  the  right  hand  object,  hold  the  sextant  upside  down, 
with  the  handle  in  the  left  hand  and  above. 

152.  Care  of  sextant. — Keep  your  sextant  in  your  own 
hands,  or  in  its  box  which  should  not  be  put  on  a  table  from 
which  it  may  be  thrown  off,  nor  in  a  roomy  drawer  wherein 
it  may  slide.     Do  not  allow  any  one  else  to  use  it.     Never 
put  it  away  damp,  as  the  moisture  will  surely  cloud  the 
mirrors  and  rust  the  metal.     Wipe  off  the  mirrors  and  arc 
with  chamois  or  silk,  but  permit  no  polishing  of  the  arc  of 
limb  or  vernier.     In  adjusting  the  instrument  when  screws 
work  against  each  other,  be  sure  to  loosen  one  as  the  other 
is  tightened.     When  in  adjustment,  do  not  tinker  with  the 
screws  even  to  remove  an  index  correction,  which,  if  small, 
can  be  allowed  for.     Keep  tangent  screw  in  mid  position. 
Keep  the  arc  clean  by  occasionally  applying  a  drop  of  ammo- 
nia, and  do  not  use  oil  except  on  screw  threads. 

153.  Resilvering  mirrors. — It  often  happens  that  mirrors 
are  injured  by  dampness  or  other  causes,  especially  when 
doing  hydrographic  work,  and  then  they  require  resilvering, 
which  may  be  done  in  the  following  way : 

Take  an  unbroken  piece  of  tin  foil  about  one-quarter  of  an 
inch  larger  in  all  dimensions  than  the  mirror  to  be  resilvered, 
lay  it  on  the  clean  surface  of  a  pane  of  glass  about  five  inches 
square,  smooth  out  the  foil,  being  careful  not  to  tear  it,  put 
a  drop  of  mercury  on  the  foil  spreading  it  carefully  with  the 


THE  ARTIFICIAL  HORIZON 


321 


finger  over  the  surface,  put  on  another  drop  and  repeat  the 
operation,  and  continue  the  process  till  the  surface  is  fluid, 
being  very  careful  that  no  mercury  gets  under  the  tin  foil. 

Lay  on  the  supporting  glass  a  piece  of  tissue  paper  so  that 
its  edge  shall  cover  the  edge  of  the  foil;  having  cleaned  the 
glass  to  be  silvered,  lay  it  on  the  tissue  paper,  and  transfer  it 
slowly  and  carefully  to  the  mercury  surface,  keeping  a  gentle 
pressure  on  the  glass  to  prevent  the  formation  of  bubbles. 


FIG.    97a. 
ARTIFICIAL  HORIZON. 

Place  the  mirror  face  downward  and  slightly  inclined,  to 
allow  any  surplus  mercury  to  run  off,  and  let  it  remain  so  till 
the  following  day,  when  the  tin  foil  should  be  trimmed  off 
flush  with  the  edge  of  the  mirror,  and  a  coating  of  varnish 
made  from  spirits  of  wine  and  red  sealing  wax  applied. 

154.  The  artificial  horizon  (see  Fig.  97a). — This  consists 
of  a  small  shallow  basin  about  8  inches  long,  4  inches  wide, 
and  f  inch  deep,  containing  mercury  or  any  other  fluid 
whose  surface  will  reflect  a  heavenly  body.  The  surface  must 


322  NAUTICAL  ASTRONOMY 

be  horizontal,  and  free  from  products  of  oxidation  or  other 
matter  diminishing  the  reflecting  power. 

The  basin  or  receptacle,  made  of  wood  or  iron,  is  covered 
by  a  roof  consisting  of  two  pieces  of  plate  glass  in  a  frame 
to  protect  the  surface  of  the  mercury  from  dust  or  wind. 

An  iron  bottle  is  furnished  to  contain  the  mercury  when 
not  in  use ;  this  bottle  is  provided  with  a  screw  stopper  and 
a  funnel  to  prevent  loss  of  mercury  in  handling ;  these  articles 
complete  the  artificial  horizon  outfit. 

In  the  absence  of  mercury,  molasses  or  oil  may  be  used; 
but,  with  oil,  the  receptacle  or  basin  must  be  blackened  on 
the  inside. 

Reflecting  horizons  of  black  glass,  plane  and  accurately 
ground,  made  level  by  levelling  screws,  are  sometimes  used, 
though  not  recommended. 

Care  of  and  preparation  of  artificial  horizons  for  use. — 
The  artificial  horizon  finds  its  principal  use  on  shore  in  rating 
chronometers,  and  then  should  be  at  its  best  and  used  under 
the  most  favorable  conditions. 

The  surface  of  the  mercury  must  be  clean  and  free  from 
dust  and  the  roof  perfectly  dry.  Scum  and  impurities  may 
be  removed  from  mercury  by  gently  drawing  over  its  surface 
the  straight  edge  of  a  piece  of  blotter  cut  to  the  length  of  the 
basin,  pressing  it  below  the  surface  of  the  mercury,  and  in- 
clining it  so  that  it  may  act  like  a  scoop.  If  the  mercury  is 
alloyed,  wash  it  with  sulphuric  acid,  then  with  water,  and 
filter  it  through  muslin. 

Before  pouring  the  mercury  into  the  basin,  see  the  basin 
well  cleaned  and  dry,  remove  the  funnel  and  stopper,  and 
screw  on  the  funnel,  so  that,  by  passing  through  a  greatly  con- 
tracted passage,  the  mercury  may  be  cleansed.  Place  a  finger 
over  the  opening,  shake  up  the  bottle,  then  invert  it,  and  let 
any  scum  rise  to  the  surface.  Hold  the  bottle  inverted  over 
the  basin,  remove  the  finger,  and  let  the  mercury  run  into  it. 


THE  ARTIFICIAL  rfoRizoN  323 

When  the  basin  is  full,  put  the  finger  over  the  aperture  and 
reverse  the  bottle;  it  is  better  to  do  this  before  all  the  mer- 
cury is  out,  or  nearly  so,  in  order  to  keep  back  any  scum  or 
impurities.  The  roof  should  be  placed  over  the  basin  for  a 
few  moments  to  allow  any  moisture  in  the  imprisoned  air  to 
be  deposited  on  the  glass  surface  of  the  roof,  which  is  then 
lifted,  wiped  off,  and  replaced.  A  piece  of  cloth  for  the  basin 
to  rest  on,  and  large  enough  to  receive  the  edges  of  the  roof, 
will  keep  out  moisture. 

A  roof  should  be  used  whose  glass  has  no  prismatic  effect, 
but  this  can  be  eliminated,  where  it  exists,  by  reversing  the 


FIG.  97b. 


roof  in  each  set  of  observations.  However,  in  observing  stars 
on  both  sides  of  the  zenith,  since  the  mean  of  results  will  be 
taken,  and  the  prismatic  effect  is  of  an  opposite  sign  in  each 
case,  the  observer  should  keep  the  same  end  of  the  roof 
towards  himself  in  each  set  of  observations. 

Advantages  of  the  artificial  horizon. — By  using  an  arti- 
ficial horizon,  and  halving  the  angle,  the  errors  of  observation, 
whether  of  instrument  or  observer,  are  also  halved ;  and  the 
correction  for  dip,  which  depends  on  both  the  height  of  eye 
and  refraction,  is  obviated,  as  the  artificial  horizon  furnishes 
a  horizontal  plane.  Its  use,  however,  is  limited  to  shore 
observations. 


324  NAUTICAL  ASTRONOMY 

Theory  of  the  artificial  horizon  (Fig.  97b). — A  ray  from 
B  is  reflected  from  the  basin  H  to  the  eye,  making  £BHH* 
=  Z.EHH"=Z.H'HB'.  A  ray  from  B  is  also  reflected  by 
the  sextant  mirrors,  making  the  sextant  image  coincident  with 
the  basin  image.  Now  £ERE'  —  £  HEB  +  Z,  HBE,  but  the 
body  B  is  so  far  off  that  the  ray  BR  is  practically  parallel  to 
the  ray  BM,  so  that  the  angle  ERE'  —  L  BEH  —  angle  read 
from  the  arc  —  2BHH',  or  twice  the  altitude  of  B. 

To  take  an  observation  of  the  sun,  using  an  artificial  hori- 
zon.-r-Select  an  observation  spot  so  that  the  basin  may  be 
evenly  placed  on  a  solid  foundation,  in  a  sheltered  position 
undisturbed  by  breezes,  or  any  movements  or  jars  in  the 
vicinity  which  might  ripple  its  surface.  In  case  equal  alti- 
tudes are  to  be  observed,  the  spot  should  be  so  selected  that, 
if  the  sun  is  observed  at  one  altitude  on  one  side  of  the  me- 
ridian, its  view  may  not  be  shut  out  by  houses,,trees,  hill  tops, 
or  other  obstructions,  when  at  the  same  altitude  on  the  other 
side  of  the  meridian. 

Clean  the  basin  and  put  it  with  its  length  nearly  in  line 
with  the  direction  of  the  sun,  but  a  little  in  advance ;  pour  the 
mercury  into  the  basin  and  see  its  surface  cleared  of  scum  and 
impurities.  Wipe  the  glass  of  the  roof  and  cover  the  basin. 

Determine  the  I.  C.  and  put  the  tangent  screw  in  mid 
position. 

The  observer  should  sit  on  a  low  stool  or  on  the  ground, 
with  his  back  supported,  if  possible ;  assuming  the  most  com- 
fortable position  possible  under  the  circumstances,  so  as  not 
to  tire  himself;  and,  at  the  same  time,  so  placing  his  eye  that 
he  may  see  the  image  of  the  sun  reflected  from  the  center 
of  the  mercury. 

Turn  down  the  necessary  shade  glasses  before  both  the 
index  and  horizon  glasses,  and,  without  putting  in  the  tele- 
scope, direct  the  line  of  sight  to  the  sun,  and  bring  it  down 
till  the  lower  limb  of  the  image  reflected  by  the  mirrors  over- 


USE  OF  ARTIFICIAL  HORIZON  325 

laps  the  upper  limb  of  the  image  reflected  by  the  mercury. 
Screw  in  the  telescope  with  a  colored  cap  on  the  eye  piece, 
throw  back  the  colored  shade  glasses,  and  proceed  to  observe 
the  altitude.  The  sextant  reading  corrected  for  I.  C.  gives 
double  the  apparent  altitude  of  the  limb  observed.  Half  the 
result,  corrected  for  S.  D.  (-(-  for  the  lower  limb,  —  for 
the  upper  limb),  parallax,  and  refraction  gives  the  true  alti- 
tude of  the  center. 

If  observing  in  the  forenoon,  the  suns  will  separate;  have 
the  time  marker  "  stand  by  "  and  as  they  separate,  the  lower 
limb  of  the  apparent  sextant  image  being  just  tangent  to 
the  upper  limb  of  the  horizon  image,  say  "  mark."  The 
assistant  notes  the  time,  records  it,  and  also  the  angle.  It 
is  usual  to  take  the  altitudes  at  equal  intervals  in  arc,  setting 
the  sextant  at  the  next  division  after  one  observation,  and 
waiting  for  tangency  or  contact.  The  interval  should  be 
sufficient  to  permit  care  and  accuracy  in  reading  and  ob- 
serving. 

If  not  afraid  of  the  prismatic  effect  of  the  shades,  different 
colored  shade  glasses  may  be  used  before  the  index  and  hori- 
zon glasses,  giving  different  colors  to  the  two  images  of  the 
sun,  and  making  it  easier  to  distinguish  them. 

If  used,  these  shade  glasses  give  sufficient  protection  to  the 
eye,  and  the  colored  cap  of  the  telescope  is  not  used. 

Determining  the  limb  observed. — If  immediately  after  con- 
tact the  two  images  of  the  sun  were  observed  to  separate  in 
the  forenoon,  or  close  in  on  each  other  in  the  afternoon,  then 
the  limb  observed  was  the  lower  limb ;  otherwise,  the  observed 
limb  was  the  upper  limb. 

To  take  an  observation  of  a  star,  using  an  artificial  hori- 
zon.— The  observer  places  himself  so  as  to  see  the  image  of 
the  star  reflected  in  the  mercury  of  the  basin ;  this  image  will 
seem  as  far  below  the  surface  of  the  mercury  as  the  real  star 
seems  above  it. 


326  NAUTICAL  ASTRONOMY 

Before  screwing  in  the  tube  or  the  inverting  telescope,  place 
the  zero  of  the  vernier  on  the  zero  of  the  limb,  direct  the  line 
of  sight  to  the  star,  both  eyes  open,  keep  the  plane  of  the  sex- 
tant vertical,  and  move  the  index  bar,  keeping  the  star's 
reflected  image  in  sight,  till  the  image  reflected  by  the  mir- 
rors (the  sextant  image)  is  in  coincidence  with  that  reflected 
by  the  mercury  and  seen  directly  through  the  center  of  the  tele- 
scope collar  and  the  horizon  glass.  The  sextant  reading  cor- 
rected for  I.  C.  gives  double  the  star's  apparent  altitude.  Half 
the  result,  or  the  apparent  altitude,  corrected  for  refraction, 
gives  the  true  altitude  of  the  star. 

If  intending  to  use  the  tube  or  inverting  telescope,  screw  it 
in  as  soon  as  the  star  has  been  brought  down,  and  proceed  with 
the  observations,  saying  "mark"  to  the  assistant  when  the 
two  images  are  in  coincidence. 


CHAPTER  X. 

CHRONOMETERS  AND  TORPEDO-BOAT  WATCHES. 
STOP  AND  COMPARING  WATCHES. 

CHRONOMETERS. 

155.  Definition. — The  term  chronometer  is  applied  to  a 
portable  timepiece  of  superior  workmanship,  furnished  with 
special  mechanism  consisting  of  compensation  balance,  bal- 
ance spring,  and  escapement,  so  constructed  as  to  obviate 
changes  in  its  rate  due  to  expansion  or  contraction  of  its 
mechanism,  through  effect  of  heat  or  cold.  The  term  chro- 
nometer, however,  is  more  generally  applied  to  one  adapted 
for  use  on  board  ship.'  A  marine  chronometer  should  beat 
half  seconds.  Its  special  function  is  to  furnish  the  time  of 
the  prime  meridian,  almost  universally  taken  as  that  of 
Greenwich. 

Mean  or  sidereal  chronometer. — A  chronometer  may  be 
regulated  to  keep  mean  or  sidereal  time ;  if  to  keep  mean  time, 
it  is  called  a  mean  time  chronometer  and  its  units  are  those 
of  mean  solar  time;  if  to  keep  sidereal  time,  it  is  called  a 
sidereal  chronometer  and  its  units  are  those  of  sidereal  time. 
(See  Art.  171.) 

The  mean  time  day  at  any  place  commences  when  the  mean 
sun  is  on  the  upper  branch  of  the  meridian  at  that  place,  and 
the  theory  of  a  mean  time  chronometer,  keeping  the  time  of 
that  place,  is  that  it  shows  Oh  Om  0s  when  the  mean  sun  is  on 
the  upper  branch  of  the  meridian. 

Practically  every  chronometer  has  an  error  and   a  daily 


328 


NAUTICAL  ASTRONOMY 


rate,  gaining  or  losing,  so  that  in  order  to  have  a  chronometer 
regulated  to  a  local  or  to  Greenwich  mean  time,  its  error  on 
that  time  must  be  known,  and  also  its  daily  rate,  i.  e.,  the 
daily  gain  or  loss.  Both  are  plus,  if  the  chronometer  is  fast 


CHRONOMETER. 

and  gaining;  minus,  if  slow  and  losing.  It  is  customary, 
however,  to  use  the  error  as  a  correction,  which  is  the  quan- 
tity to  be  applied  to  a  chronometer  reading  to  reduce  it  to 
the  correct  time  at  that  instant.  This  is  plus  when  the 
chronometer  is  slow,  and  the  daily  change  is  plus  when  the 
chronometer  is  losing.  If  the  correction  of  a  given  mean 


CHRONOMETERS  329 

time  chronometer  is  on  G.  M.  T.,  the  resulting  time  will  be 
GL  M.  T.  The  letters  C.  C.  usually  represent  this  correction. 

Since  a  sidereal  day  at  a  place  begins  when  the  vernal  equi- 
nox is  on  the  upper  branch  of  the  meridian  at  that  place,  the 
theory  of  the  sidereal  chronometer  is  that  it  should  show 
Qh  Qm  QS  when  the  vernal  equinox  is  so  situated,  and  its 
error  on  any  sidereal  time  (the  amount  it  is  fast  or  slow 
on  that  time)  and  its  daily  rate  must  be  known  in  order  to 
say  the  chronometer  is  regulated  to  that  time. 

Bating. — The  question  of  rating  a  chronometer  will  be 
considered  elsewhere  in  this  work. 

Number  for  safety. — The  U.  S.  Naval  vessels  carry  at  least 
three  good  chronometers.  In  this  way,  each  may  be  a  check  on 
the  other,  and  irregularity  on  the  part  of  one  will  be  made 
evident  by  a  comparison  of  2d  differences  as  recorded  each  day, 
it  being  assumed  that  all  will  not  have  the  same  kind  of 
daily  rate. 

There  is  no  particular  advantage  in  having  two  chronome- 
ters, except  for  the  possibility  of  injury  to  one,  for,  if  they 
begin  to  differ  widely,  after  a  period  of  regularity,  it  would 
be  difficult  to  determine  which  is  the  good  or  faulty  one,  in 
the  absence  of  means  to  check  their  indications. 

Stowage  on  board. — The  chronometer,  swung  in  gimbals  in 
its  own  case,  is  placed  in  the  chronometer  box  as  soon  as  re- 
ceived on  board.  This  box  has  as  many  compartments  as 
there  are  chronometers,  and  each  compartment  is  well  padded 
with  hair  and  lined  with  baize  cloth  to  prevent  sudden  changes 
of  temperature  and  to  reduce  shocks  and  tremors  as  much  as 
possible.  The  box  is  surrounded  by  a  strong  casing  suffi- 
ciently large  to  admit  of  a  clear  space  of  at  least  two  inches 
all  around.  Before  reception  of  the  chronometers  o*n  board, 
the  box  and  its  casing  should  be  secured  to  a  solid  block  of 
wood,  bolted  to  the  beams  of  the  deck,  in  a  part  of  the  ship 
that  is  to  be  the  permanent  abode  of  the  instruments;  as  low 


330  NAUTICAL  ASTRONOMY 

down  in  the  ship  as  possible  where  the  temperature  may  be 
equable,  and  where  gun-fire  may  have  its  least  effect;  amid- 
ships and  so  as  near  the  center  of  motion  as  convenient,  suffi- 
ciently far  forward  as  not  to  be  affected  by  vibrations  of  the 
screw;  removed  from  influences  of  masses  of  iron,  especially 
vertical  iron,  dynamos,  electric  wiring,  or  magnets  of  any 
description.  The  chronometers,  when  stowed,  should  be  al- 
lowed to  swing  freely  in  the  gimbals,  should  constantly  occupy 
the  same  relative  position,  with  the  Xll-hour  mark  towards 
the  same  part  of  the  ship,  and  should  not  be  removed  except 
for  necessity. 

Designation. — Instead  of  designating  chronometers  by  their 
numbers  or  maker's  name,  it  is  customary  to  denote  them  by 
the  letters  of  the  alphabet,  A,  B,  C,  D,  etc.  The  standard, 
called  A,  made  by  a  maker  of  well-known  reputation,  should 
have  a  first-class  record,  a  clear,  distinct  beat  to  half  seconds, 
and  a  small  stable  rate,  the  stability  of  rate  being  of  more 
importance  than  its  amount.  It  should  occupy  a  central 
position  among  the  others.  Of  course,  a  record  should  be 
made  in  the  chronometer  journal  of  the  number  and  maker 
of  chronometers  to  which  the  above  letters  may  be  assigned. 

156.  A  maximum  and  minimum  thermometer  should  be 
kept  in  the  case  with  the  chronometers,  and  recorded  at  the 
time  of  winding ;  it  should  be  kept  in  a  vertical  or  nearly  ver- 
tical position.    A  small  horse-shoe  magnet  is  used  to  reset  the 
indices.     This  magnet  should  be  kept  with  keeper  on  outside 
the  box,  and  never  brought  near  the,  chronometers.     Should 
the  mercury  column  become  divided,  move  the  indices  well 
away  from  the  column,  then  holding  the  thermometer  verti- 
cal, bulb  end  up,  give  it  a  quick  movement  downward,  bring- 
ing it  to  an  abrupt  rest.     Repeat  this  process,  if  necessary, 
till  the  break  in  the  column  disappears,  being  careful  to  keep 
the  indices  clear  of  the  mercury. 

157.  Winding. — Some  chronometers  are  made  to  run  8 


CHRONOMETERS  331 

days  and  are  wound  but  once  a  week;  however,  it  is  believed 
they  would  run  better  if  wound  every  day.  Our  service 
chronometers  will  run  for  56  hours,  and  should  be  wound 
every  24  hours,  at  regular  and  stated  times,  in  order  to  bring 
into  play  each  day  the  same  part  of  the  spring,  and  thereby 
contribute  to  regularity  and  stability  of  rate.  The  time  for 
winding  chronometers  depends  on  the  commanding  officer, 
and  may  be  just  before  8  a.  m.  or  just  before  noon;  at  all 
events,  they  should  be  reported  wound  to  the  captain  at  8  a.  m. 
or  noon,  as  the  case  may  be. 

To  wind. — Place  the  left  hand  over  the  face,  turning  slowly 
and  carefully  the  chronometer  bowl  in  its  gimbals;  then, 
holding  the  bowl  firmly  in  the  left  hand,  rotate  the  valve  with 
forefinger  or  thumb  of  left  hand,  according  to  direction  of 
rotation,  or  as  most  convenient,  until  the  key  hole  is  uncov- 
ered; place  the  key  in  position  with  right  hand,  wind  slowly 
to  the  left  the  required  number  of  half  turns,  usually  7  or  8, 
counting  the  half  turns  as  the  chronometer  is  wound,  to 
avoid  too  much  force  at  the  last  one,  though  be  careful  to 
wind  to  a  full  stop.  After  winding,  remove  the  key,  and  the 
valve  should  close  automatically;  if  it  does  not,  close  it  to 
keep  out  dust  and  dampness;  then  the  chronometer  is  eased 
to  its  original  position,  the  Xll-hour  mark  pointing  as  before. 
It  is  well  to  note  after  winding  that  the  indicator  on  face 
of  dial,  which  shows  the  number  of  hours  since  winding,  then 
reads  Oh. 

After  winding,  compare  chronometers,  fill  in  the  columns  of 
chronometer  comparison  book,  reset  maximum  and  minimum 
thermometer,  and  stow  the  magnet  away  so  as  not  to  be  any- 
where near  the  chronometers. 

\ 

When  run  down. — When  run  down,  a  chronometer  will  not 
start  on  winding;  however,  the  balance  wheel  can  be  set  in 
vibration  by  a  quick,  but  not  a  violent,  horizontal  circular 
motion. 


,°)32  NAUTICAL  ASTKONOMY 

Resetting  hands. — Should  it  be  desirable  to  reset  the  hands, 
unscrew  the  glass  cover  from  the  face,  place  the  winding  key 
on  the  projecting  stem,  and  turn  the  hands  in  one  direction 
only,  which  should  always  be  ahead.  If  practicable  to  do  so, 
avoid  altering  the  position  of  the  hands  by  starting  the  chro- 
nometer at  the  time  indicated.  Never  touch  the  hands  nor 
turn  them  backwards. 

158.  Comparison. — The  following  method  of  comparing  is 
generally  followed,  it  being  better  that  a  single  observer  should 
alone  make  the  comparison. 

The  observer  determines  upon  a  certain  time  by  standard 
for  the  comparison,  and,  holding  the  comparison  book  in  the 
left  hand,  enters  that  time  for  A.  Opening  the  glass  top  of 
A,  but  only  the  wooden  top  of  B,  so  as  not  to  hear  the  ticking 
of  the  latter,  he  takes  a  beat  from  A,  say  5  seconds,  before 
the  second  hand  is  to  reach  the  comparing  mark.  Casting 
his  eye  upon  the  dial  plate  of  B,  listening  intently  to  the 
beats  of  A,  he  counts  by  ear  the  beats  which  elapse  before  the 
second  hand  reaches  the  comparing  mark.  At  that  instant 
he  reads  the  seconds,  minutes,  and  hours  of  B.  A  second 
comparison  will  verify  the  first,  or  indicate  an  error.  Enter 
the  reading  of  B  below  that  of  A,  note  the  difference  and  2d 
differences  in  their  proper  places  in  the  comparison  book. 
Compare  C,  and  all  the  other  chronometers  in  the  same  way 
with  A,  recording  carefully  the  hours,  minutes,  and  seconds 
of  each  comparison  in  the  proper  places. 

It  is  desirable  to  consider  the  standard  fast  of  the  other 
chronometers  as  well  as  fast  on  the  Greenwich  mean  time, 
whatever  the  actual  indications  of  A  may  be,  so  that  by  sub- 
tracting the  comparison  of  any  chronometer  with  the  stand- 
ard from  the  error  of  the  standard  (adding  12  hours  to  the 
latter,  if  necessary),  the  error  of  that  particular  chronometer, 
fast  on  Greenwich  time,  is  obtained.  To  illustrate,  suppose 
A  is  fast  of  G.  M.  T.  Oh  10m  10s  and  A  —  B  =  llh  10m  50s. 


CHRONOMETERS 

The  first  may  be  written  thus      A  —  G.  M.  T.  -±     Oh  10m  10s 

and  A  —  B  =11     10     50 

Therefore, 
B  is  fast  of  G.  M.  T.  59m  20s,  or  B  —  G.  M.  T.  = ;    Oh  59m  ^n 

Comparison  of  a  mean  and  sidereal  time  chronometer. — hi 

the  comparison  of  a  mean  time  chronometer  with  a  sidereal 
time  chronometer,  the  difference  between  the  two  can  be  ob- 
tained within  a  very  small  fraction  of  a  second  by  watching 
for  the  coincidence  of  their  beats.  Since  the  second  of  sider- 
eal time  is  shorter  than  that  of  mean  time,  or  1s  of  sidereal 
time  =:  Os.99?27  mean  time,  the  sidereal  chronometer  gains 
on  the  mean  time  chronometer  Os.00273  in  1s,  and  therefore 
gains  one  beat,  or  0s. 5, 'in  183  seconds.  Hence,  once  in  about 
every  three  minutes  the  two  chronometers  beat  together,  and, 
as  the  observer,  when  watching  one,  and  counting  the  beats 
of  the  other,  fails  to  note  any  difference  in  the  beats,  he  re- 
cords the  corresponding  half  seconds  of  the  two  chronometers 
and  notes  the  minutes  and  hours  of  each. 

159.  Cleaning  and  oiling. — Chronometers  should  be  cleaned 
and  oiled  every  three  or  four  years;  when  the  oil  becomes 
dried,  or  thickened,  the  rate  will  be  irregular.     Besides,  the 
mechanism  should  be  cleaned  occasionally  to  prevent  or  re- 
move any  rust  that  may  follow  exposure  to  dampness. 

160.  Transportation. — Whenever   chronometers   are   to   be 
transported,  even  for  short  distances,  clamp  the  catch  of  gim- 
bal  ring,  and  carry  them  in  their  transportation  boxes,  or  in 
a    handkerchief    which    is    passed    under   the   box,    through 
handles,  and  square  knotted  on  top.     Great  care  must  be 
taken  to  give  them  no  shock  or  circular  movement;  when  a 
chronometer  is  carried  in  a  boat,  it  should  be  kept  suspended 
by  the  hand.     Chronometers  may  be  transported  while  run- 
ning in  the  following  way :  remove  the  bowl  from  the  gimbal 
ring,  being  careful  not  to  let  it  fall  or  be  jarred  when  un- 


334  NAUTICAL  ASTRONOMY 

screwing  the  pivot  screw  of  the  ring;  wrap  the  bowls  in  soft 
paper,  place  them,  dials  up,  in  circular  paste-board  boxes  with 
corrugated  paste-board  packing  (if  these  boxes  are  not  on 
hand,  wrap  bowls  in  cotton) ;  place  these  boxes,  tops  up, 
wrapped  in  cotton,  or  hair,  in  a  large  rectangular  basket,  as 
far  from  the  center  as  possible;  put  down  and  secure  top 
of  basket,  and  carry  it  by  its  handles,  protecting  it  from  jars 
or  jerky  movements. 

For  transportation  to  a  distance. — Chronometers  should, 
whenever  possible,  be  sent  by  hand;  but  if  necessary  to  send 
by  express,  as  when  sending  to  a  long  distance,  and  also  for 
repairs,  they  should  be  allowed  to  run  down,  be  dismounted, 
and  the  balance  stayed  with  clean  cork  at  diametrically  oppo- 
site points.  Place  the  gimbals  in  the  bottom  of  the  case, 
over  which  put  a  pad  of  cotton  wrapped  in  soft  paper  to  form 
a  seating  for  the  bowl.  The  chronometer  bowl  is  wound 
around  with  rolls  of  cotton  in  paper,  seated  on  the  cotton  pad 
in  the  bottom  of  the  case,  and  then  covered  by  a  similar  pad. 
See  the  case  tightly  packed  and  closed,  put  into  its  transport- 
ing box,  which  is  then  securely  closed  and  itself  wrapped  in 
a  thick  padded  covering,  and  marked  "  Delicate  Instrument, 
Carry  by  Strap." 

To  stay  the  balance  with  cork. — Unscrew  the  bezel,  leav- 
ing the  chronometer  movement  free  in  the  bowl.  Turn  over 
the  bowl  on  the  left  hand,  supporting  it  by  the  fingers  around 
the  dial  edge;  lift  the  bowl,  uncovering  the  movement;  stop 
the  balance  by  touching  it  very  lightly  with  a  dry  piece  of 
paper,  and  stay  it  by  two  dry  and  clean  strips  of  cork  placed 
gently  under  the  outer  rim  at  points  diametrically  opposite,  so 
as  not  to  cover  oil  holes  or  touch  any  other  parts1  of  the 
mechanism.  Replace  the  movement  in  the  bowl  and  screw 
on  the  bezel. 

161.  Effect  of  change  of  temperature. — When  marked 
changes  of  climate  are  encountered,  as  when  on  a  long  voyage, 


HARTNUP'S  LAW  335 

the  chronometer  rate  will  change,  and  this  change  is  univer- 
sally recognized  as  due  to  the  changes  of  the  temperature  ex- 
perienced by  the  chronometer;  Hartnup's  law,  governing  the 
peculiarities  developed  under  such  circumstances,  as  usually 
stated,  reads  as  follows :  "  Every  chronometer  goes  fastest  in 
some  certain  temperature,  called  the  temperature  of  compen- 
sation, and  this  can  be  calculated  for  each  chronometer  from 
rates  determined  in  three  fixed  temperatures.  As  the  tem- 
perature varies  either  side  of  this  temperature  of  compensa- 
tion, the  chronometer  goes  slower,  and  the  rate  varies  as  the 
square  of  this  variation  in  degrees." 

This  should  be  the  mean  temperature  to  which  chronome- 
ters are  subjected;  and  considering  their  actual  use,  for  navy 
chronometers,  it  is  approximately  69°  F. 

General  equation. — If  0°  be  the  temperature  of  compensa- 
tion, r  the  chronometer  rate  at  that  temperature,  z  the  tem- 
perature constant  or  change  of  rate  for  one  degree  of  tempera- 
ture either  side  of  0°,  0^°  the  temperature  for  which  the  cor- 
responding rate  is  rx ,  then  the  effect  of  temperature  alone  is 
expressed  by  the  general  equation : 

r1  =  r  +  z(0°—  O2- 

This  equation  serves  for  temperatures  between  45°  F.  and 
90°  F.,  but  outside  these  limits,  the  change  of  rate  is  propor- 
tional to  a  higher  power  than  the  square. 

The  quantities  involved  in  this  equation  differ  for  every 
chronometer.  For  the  same  chronometer  r  will  vary,  but  so 
long  as  the  temperature  compensation  is  maintained  the  same, 
that  is,  so  long  as  the  compensating  balance  is  not  changed, 
8°  and  z  will  remain  practically  constant. 

The  equation  is  that  of  a  parabola.  The  requirements  of 
the  equation  are  satisfied  by  the  general  equation  of  a  para- 
bola, 


336  NAUTICAL  ASTRONOMY 

Taking  the  axis  of  X  in  the  line  representing  the  tempera- 
ture of  compensation,  if  the  ordinate  y  is  the  variation  of 
temperature  in  degrees  from  the  temperature  of  compensation, 
the  abscissa  x  is  the  change  in  rate  for  this  variation,  then 
y  —  0°  —  0±°  and  x  =  r^  —  r;  and  for  the  value  of  y  =  unity, 


After  having  found  9°,  z,  and  r,  any  number  of  points  on 
the  parabola  may  be  found,  by  solving  the  general  equation, 
assuming  0±0  at  intervals  of  5°,  and  finding  the  correspond- 
ing r±  . 

To  find  6,  z,  and  r.  —  At  the  U.  S.  Naval  Observatory  all 
chronometers  are  subjected  to  two  tests  in  the  temperature 
room,  the  variation  being  from  90°  F.  to  50°  F.  in  the  first; 
and  from  50°  F.  to  90°  F.  .in  the  second.  The  chronometers 
are  exposed  in  each  test  for  one  week  to  certain  predetermined 
temperatures  under  certain  fixed  rules;  the  errors  being  de- 
termined at  the  beginning  and  end  of  each  week,  the  daily 
rates  for  the  several  temperatures  are  obtained  from  them. 

The  data  found  at  the  mean  temperatures  of  55°,  70°, 
and  85°  F.  are  used  in  the  general  formula  for  the  determina- 
tion of  0,  z,  and  r,  and  from  these  values,  a  curve  is  constructed 
for  each  chronometer.  This  curve  with  rates  plotted  up  to 
date,  known  as  Form  No.  1,  or  "Bate  Curve  for  Tempera- 
ture at  Observatory/'  accompanies  a  chronometer  when  issued. 
The  same  record  sheet  contains  Form  No.  2,  or  "  Rate  Curve 
and  Observations  on  Board  Ship,"  and  Form  No.  3,  or  "  Ob- 
served Errors  and  Mean  Daily  Rates  and  Temperatures." 
These,  with  Form  No.  4,  "  Record  of  Daily  Comparisons  and 
Memoranda,"  should  receive  the  navigator's  close  attention. 

162.  —  Sea  temperature  curve.  —  It  is  not  at  all  likely  that 
the  chronometer  will  have  the  same  rate  on  board  as  at  the 
observatory,  though  a  new  curve,  if  determined,  may  prove 
very  similar  to  that  found  at  the  observatory  ;  therefore,  when 
a  rate  has  been  determined  on  board,  and  is  <found  to  plot 


HACK  CHRONOMETER  337 

off  the  curve,  the  difference  between  it  and  the  curve  rate  for 
that  temperature  will  be  a  constant  to  be  applied  to  all  rates 
taken  off  the  curve. 

Any  navigator  can  compute  and  plot  the  curve  of  his  chro- 
nometers, especially  after  they  have  been  subjected  to  a  wide 
range  of  temperature.  To  do  this  let  a,  b,  and  c  each,  be  the 
mean  of  several  rates  that  differ  but  little  at  a  low,  a  mean, 
and  a  high  temperature,  and  d,  e,  and  f,  the  means  of  their 
corresponding  temperatures  ;  then,  by  substitution  in  the  gen- 

eral equation  //10       a  o\« 

rl  —  r  +  z  (0   -  -  OT_  )  2  ,  we  have 

a  =  r-\-z  (0°  —  d°)2, 


c   = 


_  _  a  —  I  _  _  _  I  —  c  _  _  _  c  —  a 

~  ~~~ 


r  =  a  —  z(8  —  dy  =  l  —  z(d  —  ef  =  c  —  z(d—f)\ 
In  the  absence  of  Form  No.  2,  take  a  sheet  of  profile  paper 
evenly  ruled  both  ways.  Let  the  horizontal  lines  represent 
degrees  numbered  at  the  left  hand,  and  the  vertical  lines 
tenths  of  seconds  numbered  at  the  top.  Take  one  vertical 
line  as  the  zero  of  rate,  depending  on  the  size  and  sign  of 
rate,  let  all  rates  to  right  be  plus  or  gaining,  to  the  left  be 
minus  or  losing.  Plot  the  points  for  every  5°,  and  trace  in 
the  curve.  The  intersection  of  this  curve  with  any  tempera- 
ture will  be  the  mean  rate  for  that  temperature  read  from 
above. 

163.  Hack  chronometer  and  comparing  watch.  —  It  has 
already  been  said  that  the  chronometers  should  not  be  subject 
even  to  occasional  removal.  This  is  true,  and  in  order  to  get 
the  chronometer  times  of  certain  desired  instants,  as  the  in- 
stant of  receipt  of  the  noon  signal,  or  the  drop  of  a  time  ball, 
etc.,  use  must  be  made  of  a  less  valuable  time  piece  known  as 


338  NAUTICAL  ASTRONOMY 

the  Hack,  an  inferior  grade  of  chronometer,  or  of  a  com- 
paring watch.  The  watch  is  in  constant  use  at  sea  for  mark- 
ing times  of  observations,  and  in  order  to  get  the  chronometer 
times  of  the  observations,  the  watch  should  be  compared  just 
before  or  just  after  (preferably  both)  with  the  standard.  In 
each  case  the  watch  reading  at  comparison  should  be  sub- 
tracted from  the  chronometer  reading  at  comparison,  adding 
12  hours  to  the  latter,  if  necessary  for  the  subtraction;  so  that 
the  difference  will  always  be  in  the  form  of  C  —  W,  and, 
therefore,  additive  to  a  given  watch  time. 

Example:   Just  before  taking  a  sight  the  watch  was  com- 
pared with  the  standard  chronometer: 

C  —  lh  30m  22s,  W  =  8h  20m  30s. 

W.  T.  of  observation  8h  31m  03s.     Find  C.  T.  of  observation. 
C  =  lh  30m  22s 
W  =  S    20     30 

C  —  W  =  5h  09m  52s 
W=8    31     03 


C.  T.  of  obs.  lh  40m  55s. 

Should  the  C  —  W  obtained  before  and  after  the  observa- 
tion, or  the  instant  for  which  the  chronometer  is  desired,  differ 
in  value,  then  the  correct  C  —  W  at  the  moment  of  observa- 
tion must  be  obtained  by  interpolation ;  as  the  total  change  in 
the  C  —  W  occurs  between  the  times  of  comparisons,  the  pro- 
portional change  from  first  comparison  to  time  of  observation 
must  be  applied  to  that  time,  or  the  first  C  —  W. 

164.  Torpedo  boat  watches. — For  use  in  torpedo  boats  the 
Department  issues,  instead  of  chronometers,  stem-winding, 
lever-escapement  watches  that  run  for  about  30  hours.  An 
extra  crystal  and  main  spring  are  provided  for  each  watch. 

Stowage. — The  watch  is  kept  in  a  wooden  inner  box  and  a 
padded  transporting  .case,  and  secured  where  it  will  be  least 
affected  by  magnetic  fields,  boat's  vibrations,  changes  of  tem- 
perature, or  moisture. 


CARE  OE  CHRONOMETERS  339 

Winding. — Watches  should  be  wound  daily  at  a  given  time, 
preferably  at  8  a.  m.,  care  being  taken  not  to  press  the  spring 
on  the  rim  near  the  stem  that  releases  the  hour  and  minute 
hands,  and  thereby  turn  them.  Should  the  hands  be  acci- 
dentally turned,  reset  the  watch  by  turning  the  hands  for- 
ward, never  backward.  In  winding  be  careful  not  to  turn  the 
stem  with  one  hand  and  the  case  with  the  other;  the  case 
should  be  firmly  held  in  one  hand,  and  the  watch  wound  by 
turning  the  stem  in  the  other  hand  with  a  careful  uniform 
motion,  coming  gradually  to  a  stop. 

General  care. — The  back  should  be  kept  closed,  avoiding 
thereby,  as  much  as  possible,  injury  to  the  works  through  dust 
and  moisture.  However,  if  a  watch  gets  wet,  it  should  be 
opened  and  drained,  then  immersed  in  alcohol  for  several  min- 
utes that  all  moisture  may  be  absorbed;  after  removal,  allow 
any  remaining  alcohol  to  drain  and  evaporate,  then  immerse 
and  keep  the  watch  in  good  lubricating  oil  till  it  can  be  sent 
for  repairs. 

The  watch  should  never  be  carried  into  a  dynamo  room  nor 
allowed  to  come  under  the  influence  of  an  electric  field. 

It  should  never  be  allowed  to  hang  freely  as  from  a  hook, 
nor  subjected  to  sudden  jerky  vibratory  motions. 

It  should  be  cleaned  and  oiled  at  least  once  in  three  years, 
and  then  only  by  an  expert  watchmaker. 

Preparations  for  shipment  by  express. — Let  the  watch 
almost  run  down,  open  the  back,  stop  the  balance  very  gently 
with  a  piece  of  tissue  paper,  and  insert  a  thin  sliver  of  cork 
or  a  piece  of  folded  tissue  paper  between  it  and  the  next 
wheel  with  the  least  possible  force ;  close  the  watch,  put  it  in 
its  box,  pack  with  cotton  in  the  transporting  case,  wrap  and 
mark  the  package,  "  Delicate  Instrument,  Handle  with  Care." 

165.  Stop  and  comparing  watches. — The  above  remarks  as 
to  winding,  care,  examination,  and  cleaning  apply  to  the  stop 
and  comparing  watches  furnished  to  a  navigator. 


CHAPTEE  XL 

COMPARISON  OF  SIDEREAL  AND  TROPICAL  YEARS.— 

THE  CIVIL  YEAR.— THE  CALENDAR.— RELATION 

OF  SOLAR  AND  SIDEREAL  TIMES. 

166.  In  considering  the  apparent  motion  of  the  sun  in  the 
ecliptic,  two  methods  are  used  in  finding  the  apparent  time 
of  its  revolution,  giving  rise  to  two  different  years.     In  the 
first  method  a  year  is  the  interval  between  two  successive  ap- 
parent passages  through  the  same  equinoctial  point,  or  the  in- 
terval between  two  successive  apparent  crossings  of  the  plane 
of  the  equator  at  the  first  point  of  Aries. 

These  particular  times  are  most  easily  and  accurately  ob- 
served at  astronomical  observatories;  and  this  period,  called 
the  tropical  or  equinoctial  year,  forms  the  basis  of  time  in  civil 
life,  since  the  changes  of  the  sun's  declination,  and,  in  conse- 
quence, the  recurrence  of  the  seasons  depend  upon  it. 

Eepeated  determinations  show  the  length  of  the  tropical 
year  to  be  365d  5h  48m  47S.8,  or  365d.2422  of  mean  solar  time. 

167.  The  calendar  and  the  civil  year. — Like  the  modern 
Mohammedan  calendar,  the  Koman  calendar  up  to  45  B.  C. 
was  based  upon  the  lunar  year  of  12  months  or  355  days. 
In  early  times,  many  of  the  religious  observances  were  con- 
nected with  the  times  of  new  and  full  moon,  and  for  this 
reason  the  priesthood  made  the  calendar  purely  lunar,  not- 
withstanding the  fact  that,  by  so  doing,  the  seasons  were 
caused  to  fall  in  different  months  in  succeeding  years,  and 
much  confusion  resulted. 


THE  CALENDAR  341 

In  45  B.  C.,  Julius  Caesar  reformed  the  calendar,  introduc- 
ing what  is  known  as  the  Julian  calendar.  Adopting  365 \ 
days  as  the  proper  length  of  the  tropical  year,  but,  recognizing 
the  importance  for  the  ordinary  purposes  of  life  of  a  year  con- 
taining an  exact  number  of  days,  he  ordered  that  the  civil 
year  should  consist  of  365  days,  except  that  in  every  fourth 
year  an  extra  day  should  be  inserted,  making  the  leap  year 
366  days;  he  also  ordered  that  the  year  should  begin  on 
January  1,  which  in  45  B.  C.  was  the  day  of  the  new  moon 
next  following  the  winter  solstice.  Up  to  that  time  the  year 
had  begun  in  March,  and  this  change  altered  the  length  of 
the  preceding  year  46  B.  C.,  and,  in  consequence,  that  year  is 
known  as  the  "  year  of  confusion." 

This  calendar  was  adopted  by  the  Council  of  Nice  in  325 
A.  D.,  in  which  year  the  vernal  equinox  fell  on  March  21. 
The  average  length  of  the  year  being  365 \  days  in  the  Julian 
calendar,  some  arrangement  should  have  been  made  to  allow 
for  the  difference  between  it  and  the  tropical  year,  which, 
however,  was  not  done. 

Owing  to  accumulated  errors,  .the  vernal  equinox  feU  on 
March  11,  in  1582,  at  which  time  Pope  Gregory  XIII  re- 
formed the  calendar  by  omitting  10  days  and  bringing  the 
vernal  equinox  back  to  March  21. 

To  guard  against  further  error  he  established  what  is 
known  as  the  Gregorian  correction,  which,  when  made,  will 
prevent  any  appreciable  error  even  in  several  thousand  years. 
Eegarding  the  year  as  365 \  days,  an  annual  error  of  about 
llm  12s  is  introduced.  This  accumulated  error  amounts  to 
about  three  days  in  400  years,  and  to  eliminate  this,  three  of 
the  inserted  days  are  to  be  left  out  every  400  years,  and  they 
are.  to  be  omitted  from  those  leap  years  completing  a  century 
not  divisible  by  400.  Thus  1700,  1800,  1900  were  not  leap 
years. 

168.  The  sidereal  year. — The  second  method  of  determin- 


342  NAUTICAL  ASTRONOMY 

ing  a  year  considers  the  interval  between  the  sun's  apparently 
leaving  and  returning  to  the  same  position  relative  to  the 
stars.  This  is  the  correct  astronomical  period  of  the  apparent 
motion  of  the  sun  through  an  arc  of  360°  of  the  ecliptic,  or  of 
one  complete  revolution  of  the  earth  around  the  sun.  This 
period  is  called  a  sidereal  year,  and  repeated  determinations 
show  the  length  of  this  year  to  be  365  days,  6  hours,  9  minutes, 
9.6  seconds  or  365d.25636  of  mean  solar  time. 

The  difference  in  length  of  the  tropical  and  sidereal  years 
is  due  to  the  precession  of  the  equinoxes,  which  causes  an 
annual  movement  of  the  first  point  of  Aries  to  the  westward 
of  50".22  of  arc. 

169.  Relation  of  solar  and  sidereal  time. — From  the  defi- 
nitions already  given  of  a  sidereal  and  a  solar  day  (Art. 
141),  and  from  a  consideration  of  the  apparent  continuous 
motion  of  the  sun  towards  the  East  in  the  ecliptic,  thus 
causing  the  sun  to  move  towards  the  West  in  the  diurnal  move- 
ment of  the  heavens  more  slowly  than  the  vernal  equinox,  it 
is  plain  that  the  solar  day  is  longer  than  the  sidereal  day ;  and 
in  the  interval  of  time  in  which  the  sun  makes  a  complete 
revolution  in  its  apparent  orbit,  that  is,  in  a  sidereal  year, 
the  mean  sun,  owing  to  its  apparent  movement  to  the  east- 
ward, falls  behind  the  stars  in  diurnal  movement  360°,  or  24 
hours,  so  that  in  a  sidereal  year,  the  number  of  daily  revolu- 
tions which  the  sun  appears  to  make  about  the  earth  is  less 
by  one  than  the  number  of  daily  revolutions  made  by  the 
vernal  equinox.  The  sidereal  year,  therefore,  which  con- 
tains 365d  6h  9m  9S.6  of  mean  solar  time,  contains  366a  6h  9m 
9s. 6  of  sidereal  time. 


CHAPTER  XII. 

TIME,— CONVERSION  OF  AEC  INTO  TIME  AND  VICE 
VERSA.— RELATION  OF  THE  L.  S.  T.,  H.  A.,  AND 
R.  A.  OF  THE  SAME  BODY.— FINDING  THE  EQUA- 
TION OF  TIME,  ASTRONOMICAL  TIME,  AND  GREEN- 
WICH TIME  AND  DATE.— GAIN  OR  LOSS  OF  TIME 
WITH  CHANGE  OF  POSITION.— CROSSING  THE 
180TH  MERIDIAN.— STANDARD  TIME. 

170.  Time  is  the  hour  angle  of  some  heavenly  body,  or  of 
a  fixed  point  in  the  heavens,  whose  apparent  diurnal  motion 
is  taken  as  a  measure.  The  earth's  motion  on  its  axis,  being 
perfectly  uniform,  furnishes  the  standard  of  measurement; 
and  hence  time  is  measured  by  the  interval  between  two  suc- 
cessive transits  of  a  heavenly  body,  or  of  some  fixed  point 
in  the  heavens,  over  the  same  branch  of  a  meridian,  this 
interval  being  called  a  day.  The  apparent  revolution  of  the 
heavens  is  due  to  the  rotation  of  the  earth  on  its  axis,  and,  as 
this  rotation  is  always  performed  in  the  same  time,  the  inter- 
val above  referred  to  would  be  the  same,  whether  measured 
by  the  apparent  motion  of  the  sun,  moon,  a  star,  or  a  fixed 
point  in  the  heavens,  were  it  not  for  the  apparent  and  real 
movements  of  these  bodies.  In  navigation  and  astronomy, 
three  kinds  of  time  are  used,  depending  on  the  celestial  point 
or  body  whose  successive  transits  over  the  same  branch  of  a 
meridian  determine  the  dav. 


344  NAUTICAL  ASTRONOMY 

171.  The  three  days  are  the  sidereal,  apparent  solar,  and 
mean  solar  days,  each  of  which  days  is  divided  into  24  hours, 
each  hour  into  60  minutes,  and  each  minute  into  60  seconds ; 
the  subdivisions  of  the  sidereal  day  being  sidereal  time,  of 
the  apparent  solar  day  apparent  time,  and  of  the  mean  solar 
day  mean  time. 

A  sidereal  day  has  already  been  defined  as  the  interval  of 
time  between  two  successive  transits  of  the  vernal  equinox, 
or  the  first  point  of  Aries,  over  the  upper  branch  of  the  same 
meridian.  The  sidereal  day  at  a  place  is  regarded  as  com- 
mencing when  that  point  is  on  the  upper  branch  of  the  me- 
ridian, and  the  sidereal  hour  angle  is  then  Oh  Om  0s.  This 
should  be  the  reading  of  sidereal  clocks  at  that  instant  when 
their  error  on  local  sidereal  time  is  zero.  -  The  sidereal  time 
at  any  instant  is  the  hour  angle  of  the  first  point  of  Aries, 
reckoned  as  already  explained  for  hour  angles. 

The  apparent  solar  day  is  the  interval  of  time  between  two 
successive  transits  of  the  true  sun  over  the  upper  branch  of 
the  same  meridian,  and  apparent  time  at  any  instant  at  a 
given  place  is  the  true  sun's  hour  angle,  reckoned  as  already 
explained  for  hour  angles.  This  is  the  time  to  which  the 
deck  clocks  at  sea  are  regulated.  Apparent  noon  is  the  in- 
stant of  the  true  sun's  upper  transit,  or  when  its  hour  angle 
is  Oh  Om  0s. 

Since  the  true  sun's  apparent  motion  is  in  the  ecliptic  and 
not  in  the  equinoctial,  and  the  motion  in  the  ecliptic  is  not 
uniform,  its  change  in  right  ascension  is  not  uniform,  and 
apparent  solar  days  are  of  unequal  length.  For  this  reason, 
apparent  time  cannot  be  kept  by  clock  mechanism,  which  re- 
quires a  standard  of  time  that  can  be  subdivided  into  unvary- 
ing lengths. 

The  sidereal  days  are  of  uniform  length,  and  sidereal  time 
is  kept  by  sidereal  clocks  at  fixed  observatories.  Owing,  how- 
ever, to  the  daily  increase  of  the  sun's  right  ascension,  the 


THE  MEAN  SUN  345 

vernal  equinoctial  point  crosses  the  meridian  approximately 
3m  56s  earlier  each  day  by  solar  time,  so  that,  whilst  the  local 
sidereal  time  -of  apparent  noon  on  March  21  is  approximately 
Oh  Om  0s,  on  September  21  it  is  approximately  12  hours,  and 
on  the  next  21st  of  March  it  is  approximately  24  hours  or  0 
hours  again.  As  solar  time  determines  the  question  of  light 
and  darkness,  which  in  turn,  regulates  the  hours  of  the  business 
world,  it  is  evident  that  sidereal  time  is  not  suited  for  the 
ordinary  and  practical  purposes  of  life,  bearing,  as  it  does,  no 
simple  relation  to  the  phenomena  of  day  and  night. 

The  mean  sun  and  mean  time. — Since  the  time  for  general 
use  must  be  uniform,  and  since  the  true  sun  is  the  body  which 
would  naturally  furnish  a  measure  of  time,  if  its  motion 
were  regular,  it  becomes  necessary  to  adopt,  instead  of  the 
true  sun,  a  fictitious  sun  called  a  mean  sun,  which  is  assumed 
to  move  in  the  plane  of  the  equinoctial  and  to  increase  its  right 
ascension  Uniformly,  that  is,  to  move  in  the  equinoctial  at  the 
mean  rate  of  the  true  sun  in  the  ecliptic,  and  the  time  meas- 
ured by  the  motions  of  this  mean  sun  is  called  mean  time. 

A  mean  solar  day  is  the  interval  between  two  successive 
transits  of  the  mean  sun  over  the  upper  branch  of  the  same 
meridian,  and  mean  solar  time  at  a  given  place  is  the  hour 
angle  of  the  mean  sun  at  that  instant.  Mean  noon  is  the 
instant  when  the  mean  sun  is  on  the  upper  branch  of  a  merid- 
ian, the  hour  angle  of  the  mean  sun  being  then  Oh  O.m  0s. 

This  is  the  time  kept  by  the  ordinary  clocks  and  watches, 
and  by  the  chronometers  carried  on  shipboard  to  give  naviga- 
tors Greenwich  mean  time. 

172.  Equation  of  time. — The  difference  at  any  instant  be- 
tween apparent  and  mean  solar  time  is  the  equation  of  time. 
It  is  also  the  difference  between  the  right  ascensions  of  the 
true  and  mean  suns;  in  other  words,  it  is  the  difference  be- 
tween the  true  sun's  right  ascension  and  mean  longitude,  the 


346 


XAUTICAL  ASTRONOMY 


true  sun's  mean  longitude  being  the  same  as  the  right  ascen- 
sion of  the  mean  sun. 


PIG.  98. 


In  Fig.  98,  a  projection  on  the  plane  of  the  horizon  and 
in  Fig.  99,  a  projection  on  the  plane  of  the  equinoctial, 

P  is  the  pole,  Z  is  the  zenith,  and  PN  is  the  meridian. 

W  is  the  West,  and  E  the  East  point  of  the  heavens. 

WNE  the  equinoctial. 

CC'  the  ecliptic. 

P  T  the  hour  circle  through  the  first  point  of  Aries. 

PA  the  hour  circle  through  the  true  sun. 

T  PA  the  right  ascension  of  the  true  sun. 

PM  the  hour  circle  of  the  mean  sun. 

°TPM  the  right  ascension  of  the  mean  sun. 

NPM  the  hour  angle  of  the  mean  sun,  or  L.  M.  T. 

NPA  the  hour  angle  of  the  true  sun,  or  L.  A.  T. 

MPA  the  equation  of  time  =  TP^T—  ^ PA— NPA— NPM. 

173.  Relation  of  local  sidereal  time,  the  hour  angle,  and 
right  ascension  of  a  given  body. — In  Fig.  98  and  Fig.  99 
PT,  PA,  PM,  and  PS  are  hour  circles  through  the  first  point 
of  Aries,  true  sun,  mean  sun,  and  a  star  (or  moon  or  planet), 
respectively ; 

^PN  equals  the  right  ascension  of  the  meridian  equals 
local  sidereal  time; 


EELATION  OF  L.  S.  T.,  H.  A.  AND  R.  A.          347 

T  PA.  equals  the  right  ascension  of  the  true  sun ; 

T  PM  equals  the  right  ascension  of  the  mean  sun ; 

VPS  equals  the  right  ascension  of  a  star  (moon  or  planet) ; 

NPA  equals  the  hour  angle  of  the  true  sun ; 

NPM  equals  the  hour  angle  of  the  mean  sun ; 

NP8  equals  the  hour  angle  of  a  star  (moon  or  planet), 

but  VPN  =  TFA  +  NPA, 

therefore,  local  sidereal  time  equals  the  right  ascension  plus 
the  hour  angle  of  the  true  sun  at  the  same  instant; 

also  T  PN  —  T  PM  +  NPM,  or 

local  sidereal  lime  equals  the  right  ascension  of  the  mean  sun 
plus  the  hour  angle  of  the  mean  sun  at  the  same  instant; 

also  T  PN  =  T  PS  +.  NPS,  or 

local  sidereal  time  equals  the  right  ascension  of  a  star  (moon 
or  planet)  plus  the  hour  angle  of  the  same  star  (moon  or 
planet)  at  the  same  instant.  In  case  any  one  of  these  bodies 
is  East  of  the  meridian  so  that  its  hour  circle  is  PM' ,  the  hour 
angle  will  be  considered  negative, 

and  T  PN  =  T  PM'  —  NPM'. 

To  state  the  case  generally,  the  local  sidereal  time  at  any 
instant  equals  the  right  ascension  plus  the  hour  angle  of  the 
same  heavenly  body  at  that  instant  (see  Art.  141). . 

In  this  connection,  we  may  now  more  clearly  define  the 
hour  angle  of  a  heavenly  body  as  the  angle  at  the  pole  between 
the  celestial  meridian  and  the  hour  circle  passing  through  the 
body,  and  which  indicates  in  hours,  minutes,  and  seconds  the 
time  elapsed  since  that  body  was  on  the  upper  branch  of  the 
meridian ;  the  length  of  said  hours,  minutes,  and  seconds,  and, 
hence,  the  duration  of  time  required  by  the  body  to  describe 
its  hour  angle  depending  on  the  day  established  by  that  body's 
diurnal  motion,  and  the  length  of  the  day  depending  upon  the 
rate  of  the  body's  real  motion,  i.  e.,  or  the  rate  of  change  of 
its  right  ascension 


348  NAUTICAL  ASTRONOMY 

Thus  the  moon,  the  planets,  the  stars,  the  true  sun,  and  the 
mean  sun,  all  have  different  rates  of  speed  in  their  apparent 
diurnal  motion,  and  while  the  hour  angle  of  any  one  of  them 
might,  for  example,  be  3  hours,  the  length  of  time  required 
for  each  to  pass  from  the  meridian  through  an  hour  angle 
of  3  hours,  measured  by  some  independent  standard,  would 
be  different  for  each  one. 

174.  Astronomical    time. — The    solar    day    (apparent    or 
mean)   is  regarded  by  astronomers  as  commencing  at  noon 
(apparent  or  mean),  when  the  sun  (apparent  or  mean)  is  at 
its  upper  culmination,  and  is  reckoned  from  0  hours  at  that 
time  to  24  hours  at  the  next  upper  culmination  of  the  same 
body.     The  day  so  considered  is  called  the  astronomical  day. 

175.  Civil  time. — The  time  used  in  the  ordinary  phases  of 
life  is  called  civil  time.     It  begins  at  midnight,  12  hours  be- 
fore the  astronomical  day  of  the  same  date,  and  is  divided  into 
two  periods  of  12  hours  each,  marked  a.  m.  and  p.  m. 

176.  Rules  for  conversion  of  civil  into  astronomical  time. — 

If  civil  time  is  p.  m.,  drop  p.  m.  and  the  hours,  minutes,  and 
seconds  will  be  those  of  the  astronomical  time  of  the  same 
date.  If  civil  time  is  a.  m.,  subtract  one  from  the  date,  add 
12  to  the  hours,  and  drop  the  a.  m. 

Examples. 

February  10,  2  p.  m.,  civil  time,  is  February  10,  2  hours, 
astronomical  time. 

.March  3,  8  a.  m.,  civil  time,  is  March  2,  20  hours,  astronom- 
ical time. 

To  convert  astronomical  time  into  civil  time. — //  the  astro- 
nomical time  is  less  than  12  hours,  it  will  be  the  civil  time 
p.  m.  of  the  same  date,  so  simply  add  p.  m. 

If  the  astronomical  time  is  greater  than  12  hours,  add  one 
to  the  date,  reject  12  hours,  and  add  a.  m. 


STANDARD  TIME  349 

Examples. 

March  21,  3  hours,  astronomical  time,  is  March  21,  3  p.  m., 
civil  time. 

November  10,  15  hours,  astronomical  time,  is  November  11, 
3  a.  m.,  civil  time. 

177.  Standard  time. — This  is  the  time  of  meridians  15° 
apart  known  as  standard  meridians ;  the  time  of  any  standard 
meridian  is  used,  for  the  convenience  of  railways  and  in  the 
business  world,  in  a  belt  of  territory  extending  as  nearly  as 
possible  7-J0  each  side  of  that  standard  meridian. 

The  standard  meridians  used  in  the  United  States  are  the 
60th,  75th,  90th,  105th,  and  120th  meridians  West  from 
Greenwich;  the  times  being  known  as  Intercolonial,  Eastern, 
Central,  Mountain,  and  Pacific,  respectively. 

To  reduce  local  mean  time  to  standard  time. — //  the  local 
meridian  is  West  of  the  standard  meridian,  add  the  differ- 
ence of  longitude  in  time  to  the  local  time;  if  the  local  me- 
ridian is  East  of  the  standard  meridian,  subtract  the  difference 
of  longitude  from  the  local  time  to  obtain  the  standard  time. 

178.  Conversion  of  arc  and  time. — The  elements  of  the 
Nautical  Almanac  are  tabulated  for  Greenwich  time   (some 
of  them  are  tabulated  for  Washington  time  also),  and  to  ob- 
tain them  for  a  given  local  instant,  the  longitude  in  time  must 
be  known.     If  expressed,  as  usual,  in  degrees  it  must  be 
properly  converted. 

Under  the  subject  of  hour  angles  and  hour  circles,  the  rela- 
tion between  arc  and  time  has  been  shown,  and  further  reflec- 
tion will  show  that,  as  the  earth  revolves  on  its  axis,  360°  of 
its  surface  as  measured  along  the  equator  pass  under  the  sun 
in  24  hours,  or  15°  in  one  hour.  Since  longitude  is  measured 


350  NAUTICAL  ASTRONOMY 

along  the  equator  it  may  be  expressed  in  arc  or  time,  the  re- 
lation being: 

15°  of  arc  =  1  hour  of  time  or  1°  of  arc  =  4  minutes  of 
time. 

15'  of  arc  =  1  minute  of  time  or  1'  of  arc  =  4  seconds  of 
time. 

15"  of  arc  =  1  second  of  time  or  1"  of  arc  =  iV  second  of 
time. 

If  X  be  a  given  number  of  degrees  or  minutes  of  arc,  then, 

v 

remembering  that  „    (result  in  hours  or  minutes)   will  be 

j£- 

-=  X  60  =  X  X  4  (result,  respectively,  in  minutes  or  seconds 
15 

depending  on  whether  X  is  in  degrees  or  minutes),  any  num- 
ber of  degrees  or  minutes  not  exactly  divisible  by  15  may  be 
reduced  to  the  lower  denomination  in  time  by  multiplying  by 
4,  and  the  reverse  also  holds  true. 
Hence,  to  convert  arc  into  time: 

(1)  Divide  the  degrees  of  arc  by  15;  the  result  will  be 
hours. 

(2)  Multiply  remaining  degrees,  if  any,  by  4;  the  result 
will  be  minutes  of  time. 

(3)  Divide  minutes  of  arc  by  15;  the  result  will  be  min- 
utes of  time,  to  be  added  to  minutes  of  time  obtained  by 
rule  (2). 

(4)  Multiply  remaining  minutes  of  arc  by  4;  the  result 
will  be  seconds  of  time. 

(5)  Divide  seconds  of  arc  by  15;  the  result  will  be  seconds 
.and  decimals  of  a  second  of. time,  to  be  added  to  seconds  of 

time  obtained  by  rule  (4). 
To  convert  time  into  arc: 

(1)  Multiply  the  hours  by  15;  the  result  will  be  degrees. 

(2)  Divide  the  minutes  of  time  by  4;  the  result  will  be 
degrees,  to  be  added  to  the  degrees  obtained  by  rule  (1). 


ARC  AND  TIME  351 

(3)  Multiply  the  remaining  minutes  by  15;  the  result  will 
be  minutes  of  arc. 

(4)  Divide  seconds  of  time  by  4;  the  result  will  be  min- 
utes of  arc,  to  be  added  to  the  minutes  of  arc  obtained  by 
rule  (3). 

(5)  Multiply  remaining  seconds  and  decimals  by  15;  the 
result  will  be  seconds  of  arc. 

Ex.  72.— Convert  111°  32'  40"  into  time. 

111° 

f^L  =  7h  with  remainder  6  =  7h  24m  00s 

15 
09' 

J±.    =  2m  with  remainder  2  =     •     2     08 
15 

40" 


Therefore,  111°  32'  40"  =  7h  26m  10". 67 

Ex.  73.— Convert  9h  58m  148.5  into  arc. 

9h  X  15  =  135°  00'  00" 

51-  =  14°  with  remainder  2     ='    14     30    00 
4 

i!?=    3'   with  remainder  2.5  =  3    37.5 


Therefore,  9h  58m  14P.5  =149°  33'  37".5 

Table  7  of  Bowditch  gives  the  inter-conversion  by  inspec- 
tion. 

Examples. — Convert  into  arc. 

(74)  6h  15m  32s  Ans.     93°  53'  00" 

(75)  10    53     45  Ans.  163     26    15 
(78)   11    35     13  Ans.  173     48    15 

Convert  into  time. 

(77)  29°  43'  30"  Ans.     lh  58m  54s 

(78)  155  13  43        Ans.  10  20  54.87 

(79)  177  15  30       Ans.  11  49  02 


352 


NAUTICAL  ASTRONOMY 


179.  Relation  between  the  local  times  at  two  places. — In 

comparing  corresponding  times  at  different  meridians,  since 
the  hour  angles  increase  positively  to  westward,  the  most 
easterly  meridian  is  that  at  which  the  hour  angle  of  the  body, 
or  the  time,  is  the  greatest ;  and,  since  the  longitude  of  a  place 
is  the  inclination  of  the  meridian  of  that  place  to  that  of 
Greenwich,  it  is  the  Greenwich  hour  angle  of  a  body,  when  on 
the  local  meridian:  Therefore,  if  the  longitude  be  added  to 
the  hour  angle  of  a  heavenly  body  at  a  given  place,  the  result 
will  be  that  body's  hour  angle  from  Greenwich. 

Let  AX  and  A2  be  the  longitudes  of  two  places ;  t±  and  t2 , 
the  hour  angles  at  the  two  places  of  one  and  the  same  body 
or  point,  as  for  instance  the  vernal  equinox,  of  the  true  or 
mean  sun,  and  hence  the  local  sidereal  time,  local  apparent 
or  mean  time.  Then,  using  the  quantities  at  the  first  me- 
ridian, 

the  Greenwich  sidereal,  apparent,  or  mean  time  =  At  -f-  t^ ; 
and  using  the  quantities  at  the  second  meridian, 

the  Greenwich  sidereal,  apparent,  or  mean  time  =  A2  +  t2 . 

Therefore,  A±  +  ^  =  A2  +  t2  and  A2  —  Aj  =  t:  —  t2 , 

or  the  difference  between  the  local  times  at  two  places  equals 

the  difference  of  their  longitudes, 
the  times  being  sidereal  times,  ap- 
parent times,  or  mean  times. 
This  is  shown  graphically  in  Fig. 
100. 

Let  PM  and  PM'  be  the  merid- 
ians of  two  places,  and  MPM'  the 
difference  of  longitudes  of  the  two 
meridians ;  PS,  the  hour  circle  of  the  sun  (apparent  or  mean)  ; 
FT,  the  equinoctial  colure  (an  hour  circle) ;  then  MPS  is  the 
hour  angle  of  the  sun  (apparent  or  mean)  at  all  places  on  the 
meridian  PM ,  M'PS  is  the  hour  angle  of  the  same  body  at  all 
places  on  the  meridian  PM',  JlfPT  and  M'PY  are  the  corre- 


To  FIND  GREENWICH  DATE  353 

spending  sidereal  times  at  the  two  meridians,  and  MPM' 
=  MP8  —  M'PS  =  MPV  —  M'P<r  ;  or  the  difference  of 
the  times  at  two  places,  whether  apparent,  mean,  or  sidereal 
times,  equals  the  difference  of  longitudes  of  the  two  places. 

If  PM  be  the  meridian  of  Greenwich,  or  the  origin  of  longi- 
tudes, the  difference  of  longitude  of  Greenwich  and  places  on 
the  meridian  PM'  will  be  the  longitude  of  those  places  from 
Greenwich,  in  other  words,  MPM'  is  the  longitude  of  the  me- 
ridian PM' ;  MPS  becomes  the  Greenwich  time  (apparent  or 
mean),  and  MPT  becomes  Greenwich  sidereal  time,  and 
MPM'  =  MPS  —  M'PS  =  M P  T  —  M'P  T  ;  or  the  longitude 
of  any  meridian  equals  the  difference  between  the  time  at 
that  meridian  and  the  Greenwich  time  for  the  same  instant. 

If  G.  T.  represents  Greenwich  time,  and  L.  T.  represents 
local  time  at  a  place  in  longitude  A,  then,  if  A  is  West  longi- 
tude, the  meridian  of  Greenwich  is  to  the  eastward  of  the  local 
meridian  and  G.  T.  is  greater  than  L.  T. ;  therefore, 
G.  T.  =  L.  T.  +  A, 

G.  T.  and  L.  T.  being  reckoned  astronomically  from  0  hours  to 
24  hours,  to  the  westward.  However,  if  A  is  East  longitude, 
the  local  meridian  is  to  the  eastward  of  the  Greenwich 
meridian,  the  local  time  is  greater  than  the  corresponding 
Greenwich  time,  and 

G.  T.  =  L.  T.  —  A. 

Hence  the  following  rules  in  finding  the  Greenwich  time 
and  date: 

(1)  Express  the  local  or  ship's  date  and  time  astronomically 
in  days,  hours,  minutes,  and  seconds. 

(2)  To  this  local  astronomical  date  and  time  add  the  longi- 
tude in  time,  if  West,  and  the  sum  in  days,  hours,  minutes, 
and  seconds  will  be  the  Greenwich  astronomical  date  and  time. 


354  XAUTICAL  ASTRONOMY 

(3)  //  in  East  longitude,  subtract  the  longitude  in  time 
from  the  local  astronomical  time  expressed  in  days,  liours, 
minutes,  and  seconds,  and  the  result  will  ~be  the  Greenwich 
astronomical  date  and  time. 

EX.  80.— In  Long.  76°  26'  Wv  the  local  time  being  8h  09m 
41s  p.  m.,  April  3,  find  the  Greenwich  time. 

d     h     m      s 

Local  astronomical  time  April  3,     8    09    41 

Longitude  in  time  West  +      5    05    44 

Greenwich  astronomical  time  April  8,  13   15    25 

Ex.  81.— In  Long.  70°  22'  00"  W.,  the  local  time  being 
gh  4gm  iys  a>  m>^  january  6,  find  the  Greenwich  time. 

d     h    m     s 

Local  astronomical  time  Jan.  5,  20    48    17 

Longitude  in  time  West  +     4   41    28 

Greenwich  astronomical  time  Jan.  6,     1    29    45 

Ex.  82.— In  Long.  103°  58'  E.,  the  local  time  being  Febru- 
ary 15,  7h  35m  40s  a.  m.,  find  the  Greenwich  time. 

d     h     m     s 

Local  astronomical  time  Feb.  14,  19    85    40 

Longitude  in  time  East  (  — )  6    55    52 

Greenwich  astronomical  time  Feb.  14,  12    39    48 

Ex.  83.— In  Long.  135°  15'  E.,  the  local  time  being  Jan- 
uary 20,  5h  10m  30s  p.  m.,  find  the  Greenwich  time. 

d     h     m      s 

Local  astronomical  time  Jan.  20,     5    10    30 

Longitude  in  time  East  (— )   9    01    00 

Greenwich  astronomical  time  Jan.  19,  20   09   30 

By  pursuing  a  course  just  the  reverse  of  the  above,  sub- 
tracting West  longitudes  from/  and  adding  East  longitudes  to 
given  Greenwich  times  expressed  astronomically,  the  local 


To  FIND  GREENWICH  I)ATE  355 

astronomical  times  may  be  found  and  converted  into  local 
civil  times. 

180.  To  find  the  Greenwich  date  and  mean  time  from  the 
time  data  of  an  observation. — Navigators  are  supplied  with 
chronometers  from  which  to  obtain  the  Greenwich  mean 
time  of  their  observations,  and  for  this  time  the  various  ele- 
ments involved,  such  as  declination,  right  ascension,  etc.,  are 
taken  from  the  Nautical  Almanac. 

Having  the  error  on  G.  M.  T.  and  the  daily  rate  before 
leaving  port,  the  error  of  the  chronometer  at  any  given  in- 
stant can  be  found. 

To  the  watch  time  of  an  observation  add  the  C — W,  or  the 
difference  between  the  chronometer  and  watch,  obtained  by 
comparison  just  before  or  after  the  observation,  to  get  the 
corresponding  chronometer  time.  To  this  chronometer  time, 
apply  its  error  on  Greenwich  mean  time,  adding  if  the  chro- 
nometer is  slow,  subtracting  if  it  is  fast.  The  result,  or  the 
result  plus  12  hours,  will  be  the  G.  M.  T.  This  ambiguity 
in  the  number  of  hours  arises  from  the  fact  that  chronometer 
dial  plates  are  graduated  like  watches  from  0  hours  to  12 
hours,  instead  of  from  0  hours  to  24  hours,  and  it  is  neces- 
sary to  know  whether  the  chronometer  is  a.  m.  or  p.  m.  in 
order  to  fix  the  true  Greenwich  time  and  date.  However, 
there  need  be  no  trouble  or  ambiguity,  if  approximate  Green- 
wich time  and  date  are  gotten  from  the  approximate  local 
time  by  the  rules  just  given  in  Art.  179,  and  compared  with 
the  result  obtained  as  above  from  the  watch  time  and  chro- 
nometer comparison. 

The  following  examples  will  show  how  to  decide  whether 
the  G.  M.  T.,  in  a  given  case,  is  the  corrected  chronometer  face 
or  this  quantity  plus  12  hours. 


356  NAUTICAL  ASTRONOMY 

Ex.  84- — October  31,  about  5  a.  m.,  the  time  data  of  an 
observation  were,  W.  7h  25m  12s,  C— W  lh  44m  17s,  chronom- 
eter fast  on  G.  M.  T,  27m  31s,  Long.  8h  E.  Find  G.  M.  T. 

!h  m  a 
W.             7  25  12 
C  — W           1  44  17 

C.              9  09  29 
C.  C.        (  — )  27  31 
Greenwich  ast.  time  Oct.   30,  9  approx.  I 

[G.  M.  T.  Oct.  30,  8  41  58 

These  results  are  so  close  as  to  remove  all  doubt. 

Ex.  85. — November  17,  about  10h  53m  a.  m.,  in  Long. 
2h  20m  10s  E.,  the  time  data  of  an  observation  were  W.  4h  15m 
27s,  C— W  4h  07m  20s,  chronometer  slow  on  G.  M.  T.  10m  15s. 
Find  G.  M.  T. 


Civil  time  Nov.  17 


d     h     m     s 


h   m    s 

W.  4  15  27 

C-W  4  07  20 


(a.m.)  j  °53       aPPrOX' 

Local  ast.  time  Nov.     16,  22  53       approx. 
Long.  East  (-)  2  20  10  |£  8  23  47 

Greenwich  ast.  )  Noy    16    20  32  50anDrox    '  °'  °' 

time.  J  I  G.  M.  T.  Nov.  16,     20  33  02 

It  will  be  noticed  that  in  the  above  example  12  hours  must 
be  added  to  the  chronometer  time  and  the  date  of  the  pre- 
vious day  taken  to  make  the  two  Greenwich  times  and  dates 
more  nearly  agree. 

It  may  sometimes  be  the  case  that  the  watch  by  which  ob- 
servations are  taken  is  not  regulated  to  local  time,  in  fact, 
may  be  far  out,  and  that  the  approximate  time  of  observation 
is  known  no  closer  than  by  the  words  a.  m.  or  p.  m.  (as  in 
the  case  of  set  examples).  But  the  result  may  be  reasoned 
out  correctly  thus :  Apply  the  C— W  and  C.  C.  to  the  W.  T. 
of  observation.  The  corrected  chronometer  reading,  or  this 
reading  plus  12  hours,  will  be  the  required  G.  M.  T. 

To*  determine  which  proceed  thus :  Express  the  given  ap- 
proximate local  civil  time  astronomically  and  find  the  as- 


To  FIND  GREENWICH  DATE 


357 


tronomical  date  and  hour-limits  between  which  the  true  local 
time  should  lie,  these  limits  being  determined  by  the  words 
a.  m.  or  p.  m. ;  thus  to  illustrate,  for  Sept,  15,  a.  m.,  the  limits 
of  astronomical  date  and  time  would  be  Sept,  1-1,  12h  to  24h, 
and  for  Oct.  10,  p.  m.,  they  would  be  Oct.  10,  Oh  to  12h. 

Apply  the  longitude  (adding  if  West,  subtracting  if  East) 
to  the  local  astronomical  date  and  hour-limits,  and  the  result 
will  be  the  Greenwich  astronomical  date  and  hour-limits  be- 
tween which  the  true  Greenwich  time  must  lie. 

"  Then,  if  the  corrected  chronometer  reading  falls  between 
the  Greenwich  limits,  it  is  the  correct  G.  M.  T.;  if  not,  add 
12h  to  the  corrected  chronometer  reading  and  the  result,  fall- 
ing between  the  limits,  will  be  the  G.  M.  T.,  the  date  in  either 
case  being  that  of  which  the  hours  are  a  part. 

Ex.  86.— November  15,  a.  m.  time,  in  Long.  10h  W.,  W.  T. 
of  an  observation  was  7h  50m  30s,  C— W  6h  10m  20s,  C.  C. 
+  4m  20s.  Find  the  G.  M.  T. 


h   m    8 

W.  7  50  30 

C-W  6  10  20 

C.  2  00  50- 

C.  C.  +4  20 

Corrected 

chro.         J.2   05  10 

reading. 


Approximate  local  civil  time  Nov.  15,  a.  m. 
Local  astronomical  date 

and  hour  limits. 
Longitude  West  +'10  10 

Greenwich  astronomical 

date  and  hour  limits. 


Nov.  14d  22h  to  15''  10»» 


The  corrected  chronometer  reading  falls  between 
the  limits,  the  hours  are  of  the  15th,  .-.  G.  M.  T. 
Is  Nov.  15,  2"  05m  108. 


Ex.  86 (a). — July  11,  p.  m.  time,  in  Long.  150°  E.,  the  time 
data  of  an  observation  were  as  follows:  W.  =  9h  12m  17s, 
C— W  8h  15m  14s,  C.  C.  +  10m  20s.  Find  G.  M.  T. 


h  m    s 

W.  9  12  17* 

C— W  8   15  14 

C.  5  27  31 

C.  C.  +  10  20 

Corrected 

chro.         V  5  37  51 

reading. 


Approximate  local  civil  time  July  11,  p.  m. 
Local  astronomical  date 
and  hour  limits. 


Longitude  East 
Greenwich  astronomical 
date  and  hour  limits, 


}   July 
al  }    July 


lld    Oh  to 


—  10 


12»> 
10 


Uh   to  lld  21' 


358 


NAUTICAL  ASTRONOMY 


The  corrected  chronometer  reading  does  not  fall  between  the 
above  limits,  so  adding  12h  to  it  gives  l?h  37m  51s.  which  falls 
between  the  limits;  the  hours  are  a  part  of  the  10th;  and  .'. 
G.  M.  T.  =  July  10,  17h  37m  51s. 

In  case  the  local  time  is  given  as  an  exact  time  and  the 
longitude  as  merely  East  or  West,  the  same  method  holds  good. 

Ex.  S7. — Jan.  10,  8  a.  m.,  in  West  longitude,  given  the  fol- 
lowing data:  W.  T.  9h  40m  30s,  C— W  5h  20ra  20s,  C.  C. 
+  5m  10s.  Find  G.  M.  T. 


W. 
c-w 


h  m  s 
9  40  30 
5  20  20 


C  3  00  50 

C.  C.  +  5  10 

Corrected  "V 

chro.         l3  06  00 

reading,  j 


Local  civil  time  Jan.  10,  8  a.  m. 
Local  astronomical  time       Jan.      9rt  20h 
-Longitude  West  +          0      to 

Greenwich  astronomical  ) 
date  and  hour  limits,    j 


Jan.       9<i  20>>  to  10'1    8>» 


The  corrected  chronometer  reading  falls  between  the  limits, 
the  hours  being  of  the  10th.  .'.G.  M.  T.  —  Jan.  10.  3h  06m  00s. 

If  both  time  and  longitude  are  given  as  within  a  twelve-hour 
range ;  i.  e.,  time  only  as  a.  m.  or  p.  m.  and  longitude  only  as 
E.  or  W.,  the  limits  of  the  approximate  Greenwich  astronom- 
ical time  will  be  24  hours  apart  and  two  solutions  will  result. 

Ex.  87 (a). — Aug.  10,  a.  m.  time,  in  West  Long.,  given  the 
following  data :  W.  T.  8h  00m  10s,  C— W  6h  40m  20s,  C.  C. 
+  5m  30s.  Find  G.  M.  T. 


h  m  s 
8  00  10 
6  40  20 


W. 
C— W 

C 

C.  C. 
Corrected  ) 

chro.         I  2  46  00 

reading,  j 


2  40  30 
+      5  30 


Approximate  local  civil  tinle  Aug.  10,  a.  m. 

dale 


Longitude  West 
Greenwich  astronomical 
date  and  hour  limits 


+    Oi>  —  12»>        Oh—  12>> 


al  |  . 
.    }       S" 


12HolOd 


GAIN  OR  Loss  OF  TIME  350 

Here  both  the  corrected  chronometer  reading  and  that  read- 
ing -f  12  hours  fall  between  the  limits,  and  hence  we  have  a 
double  solution : 

G.  M.  T.  =  Aug.     9,  14h  46m  00s, 
=  Aug.  10,     2h  46ra  00s. 


Examples  for  practice.     Find  G.  M.  T.,  given, 


W.  T. 
of  Obs. 

C—  W 

C.  C. 

Approx. 
Local  date. 

Long. 

Answers. 

h  m    s 

h  m    s 

m    s 

0 

h  m     s 

(88) 

3  23  43 

6  11  33 

(—  )  7  18 

April  23,   a.  m. 

90  W 

April  22,  21  27  58 

(89) 

7  53.26 

4  38  56 

(—  )  9  27.6 

Jan.    19,  a.  m. 

30  W 

Jan.     19,     0  22  54.4 

(90) 

11  49  33 

3  59  30 

(—  )30  22 

Oct.    22,   a.  m. 

120  E 

Oct.    21,  15  18  41 

(91) 

5  20  21 

3  16  24 

(—  )25  21 

Nov.     5,  p.  m. 

135  E 

Nov.     4,  20  11  24 

(03) 

7  35  10 

10  20  17 

+  10  04 

Nov.  9,  8  a.  m. 

30  E 

Nov.     8,  18  05  31 

181.  Gain  or  loss  of  time  with  change  of  position. — Cross- 
ing the  180th  meridian. — It  has  been  shown  that  local  noon 
at  any  meridian  is  when  the  sun  is  on  the  upper  branch  of  that 
meridian,  that  at  all  places  East  of  that  meridian  at  that  in- 
stant it  is  past  noon,  or  time  is  fast  of  that  of  the  given  merid- 
ian ;  at  all  places  West  of  that  meridian  it  is  not  yet  noon,  or 
time  is  slow  of  that  of  the  given  meridian.  Hence  it  is  evident 
that  if  a  navigator  travels  East,  carrying  a  watch  regulated  to 
the  time  of  the  meridian  departed  from,  and  if  he  desires  to  set 
the  watch  to  the  time  of  a  meridian  to  the  eastward,  he  must 
set  it  ahead  at  the  rate  of  1  hour  for  15°  change  of  longitude, 
or  24  hours  for  every  360° ;  in  other  words,  in  going  east- 
ward around  the  world,  or  through  360°  of  longitude  meas- 
ured in  an  easterly  direction,  he  gains  a  day  in  his  reckoning 
of  time. 


360  NAUTICAL  ASTRONOMY 

In  the  same  way,  if  sailing  westward  around  the  world, 
or  through  360°  of  longitude  measured  in  a  westerly  direc- 
tion, he  loses  a  day  in  his  reckoning  of  time. 

So  that  if  he  leaves  the  given  meridian  and  goes  around 
to  the  eastward,  keeping  his  time  regulated  to  each  succes- 
sive local  meridian,  his  reckoning  of  time  on  return  to  his 
point  of  departure  will  be  one  day  ahead  of  the  local  reckon- 
ing ;  in  other  words,  he  would  think  it,  say,  Thursday  when  in 
reality  it  was  Wednesday. 

Had  he  gone  around  to  the  westward,  he  would  have  logged 
his  return  as  Tuesday,  if  the  day  in  reality  was  a  Wednesday. 

To  avoid  such  misconceptions  and  to  keep  accurate  run  of 
dates,  when  crossing  the  meridian  of  180°,  going  eastward, 
repeat  one  day;  when  crossing  it,  going  westward,  drop  one 
day  from  the  calendar;  at  the  same  time  changing  the  name 
of  the  longitude. 

Illustration. — Suppose  a  ship,  going  eastward,  crosses  the 
180°  meridian  at  local  apparent  noon,  April  10;  find  the 
corresponding  Greenwich  time  and  date.  Then,  from  this 
result,  considering  the  longitude  as  of  opposite  name  to  that 
first  used,  find  the  local  time  and  date. 

h       m       a 

Local  apparent  time  00  00  00  April  10 

Longitude  180°  East  12  00  00  East 


Greenwich  apparent  time  12  00  00  April  9 

Longitude  180°  West  12  00  00  West 

Local  apparent  time  00  00  00  April  9 

In  other  words,  going  eastward,  and   crossing  the   180 ( 
meridian,  repeat  a  day. 


361 


Suppose  the  conditions  of  the  illustration  to  be  as 
except  the  ship  is  going  westward,  and  the  chango 
from  West  to  East  longitude,  then, 

h       m       a 

Local  apparent  time  00  00  00  April  10 

Longitude  180°  West  12  00  00  West 


Greenwich  apparent  time  12  00  00  April  10 

Longitude  180°  East  12  00  00  East 

Local  apparent  time  00  00  00  April  11 

In  other  words,  going  westward  and  crossing  the   180 ( 
meridian,  omit  one  day  from  the  calendar. 


CHAPTEE  XIII. 

NAUTICAL  ALMANAC   AND   SUBOKDINATE   COMPUTA- 
TIONS. 

182.  The  Ephemeris  and  Nautical  Almanac*  published  by 
authority  of  Congress  is  subdivided  into  three  general  parts, 
but  all  the  information  required  by  navigators  is  contained  on 
pages  2-177,  233-510,  and  650-697  (edition  of  1912). 

Pages  2-145  of  Part  I  comprise  what  is  called  the  Calendar 
giving  the  data  under  the  heads  of  the  several  months;  each 
month  has  assigned  to  it  12  pages  numbered  by  the  Roman 
numerals  I  to  XII.  In  this  book  reference  will  be  made  only 
to  the  contents  of  those  pages  of  use  to  the  navigator. 

Page  I  gives  for  Greenwich  apparent  noon  of  each  day  the 
sun's  apparent  right  ascension,  declination,  and  equation  of 
time,  with  hourly  differences  in  adjoining  columns.  The 
hourly  differences  themselves  are  for  the  instant  of  apparent 
noon  at  Greenwich,  and,  when  great  accuracy  is  required,  cor- 
rections should  be  made  for  second  differences.  The  sun's 
semi-diameter  and  the  sidereal  time  of  the  semi-diameter  pass- 
ing the  meridian  are  also  given.  The  chief  use  of  page  I  is 
for  observations  when  the  sun  is  on  the  meridian,  as  for  lati- 
tude, in  which  case  longitude  is  the  Greenwich  apparent  time. 

Page  II  contains  for  Greenwich  mean  noon  of  each  day  the 
sun's  right  ascension,  declination,  equation  of  time,  and  side- 
real time  of  mean  noon  (R.  A.  M.  O )  with  their  hourly  dif- 
ferences. Where  great  accuracy  is  desired  second  differences 
should  be  used. 

Page  IV  contains  the  moon's  semi-diameter  and  equatorial 

*  A  Nautical  Almanac,  also  issued,  contains  the  Calendar,  the  geocentric  ephem- 
erides  of  the  five  visible  planets,  mean  places  of  200  fixed  stars,  a  monthly  list  of 
stars  for  navigators,  tables  giving  approximate  times  of  meridian  transit  for  certain 
stars,  *  and  h  of  stars  on  P.  V.,  a  star  map,  and  Tables  I  to  VII  of  the  "  Ephemeris 
and  Nautical  Almanac." 


NAUTICAL  ALMANAC  363 

horizontal  parallax  for  each  mean  noon  and  midnight  of 
Greenwich  mean  time  and  the  hourly  changes  of  the  horizontal 
parallax.  The  mean  time  of  the  moon's  upper  transit  to 
tenths  of  a  minute  and  the  moon's  age  are  given  on  this  page. 

Pages  V  to  XII  contain  the  moon's  right  ascension  and 
declination  for  each  day  and  hour  of  Greenwich  mean  time 
with  the  differences  for  one  minute  opposite  each  hour. 

Pages  146-177  contain  the  right  ascensions  and  declinations 
of  the  seven  major  planets  for  the  instant  of  Greenwich  mean 
noon;  also  their  times  of  meridian  passage  at  Greenwich  to 
tenths  of  a  minute. 

Pages  233-486  give  the  right  ascension  and  declination  of 
825  principal  stars,  their  mean  places  being  on  pp.  233-250. 

Signs. — In  the  Nautical  Almanac  a  +  sign  before  the 
hourly  difference  of  declination  means  the  heavenly  body  is 
moving  toward  the  North ;  a  —  sign,  under  like  circumstances, 
means  the  body  is  moving  toward  the  South ;  a  -f-  sign  before 
the  declination  of  a  planet  or  star  means  North,  a  —  sign 
means  South ;  in  all  other  cases  +  means  increasing,  —  means 
diminishing  numerically. 

In  the  examples  of  this  book,  declinations  will  be  charac- 
terized by  the  letter  N.  when  North,  by  letter  S.  when  South ; 
and  hourly  differences  by  letter  N.  if  the  body  is  moving 
toward  the  North,  otherwise  by  the  letter  S. 

183.  Greenwich  time  essential. — Before  using  the  Almanac, 
the  correct  Greenwich  time  must  be  obtained,  as  the  elements 
likely  to  be  used  are  tabulated  for  that  time,  except  in  the 
cases  of  the  apparent  places  of  fixed  stars  which  are  tabulated 
for  Washington  time ;  so,  when  taking  out  the  right  ascension 
and  declination  of  fixed  stars  for  a  given  instant,  first  find  the 
Washington  time  corresponding. 

Work  from  nearer  noon. — Good  judgment  tells  us  to  work 
always  from  the  nearer  Greenwich  noon,  and,  if  the  Green- 
wich time  of  a  given  date  is  greater  than  12  hours,  it  is  better 


364  NAUTICAL  ASTRONOMY 

to  regard  it  as  less  than  12  hours,  and  a  minus  quantity  of 
the  next  day.  For  example,  if  G.  M.  T.  =  17*.  3  March  3, 
regard  it  as  (—  )  6h.7  March  4,  and  proceed  accordingly, 
working  backwards. 

184.  Second  differences.  —  Ordinarily  first  differences  will 
do  in  taking  out  the  various  elements  •  in  such  cases  change  in 
the  element  is  regarded  as  proportional  to  the  small  intervals 
of  time  employed;  but,  if  great  precision  is  required,  as  in 
taking  out  the  sun's  declination  and  equation  of  time  for  use 
in  equal  altitudes  for  chronometer  error,  the  reduction  for 
second  differences  should  be  introduced,  and  the  hourly  differ- 
ence interpolated  for  the  middle  instant  between  the  Almanac 
date  and  the  given  time,  thus  using  the  mean  rate  of  change 
during  the  interval.  However,  it  must  not  be  forgotten  that 
the  quantities  in  the  Almanac  are  approximate,  given  only  to 
a  certain  decimal,  and  that  it  is  useless  to  interpolate  to  a  lower 
order  than  said  decimal.  Besides,  at  sea,  where  the  time  may 
be  in  error,  excess  of  refinement  in  making  corrections  will 
not  contribute  to  accuracy. 

As  all  the  examples  of  this  work  are  meant  to  be  practical, 
second  differences  will  be  used  only  in  the  cases  of  the  sun's 
declination  and  equation  of  time  as  referred  to  above. 

Letting  H^  be  the  hourly  difference  for  the  Greenwich  noon 

preceding  the  given  Greenwich  time, 
H2  be  that  for  the  following  noon, 
t  be  the  number  of  hours  of   Greenwich  time 
after  the  first  date,  for  which  the  value  of  the 
quantity  is  required, 

TT    _        rr  4- 

then  #!  rb  —  2cM      •   X       will  be  the  mean  hourly  difference. 


In  case  t  equals  the  hours  of  Greenwich  time  before  the  second 
date,  and  the  value  of  the  quantity  is  required  for  that  instant, 

then  HI  ^F      CM       x  ^r  w^  ^e  ^ne  mean  hourly  difference. 


NAUTICAL  ALMANAC 


365 


EXf  g$f  —  On  April  2,  1905,  the  hourly  difference  of  declina- 
tion of  the  sun  at  Greenwich  apparent  noon  was  X.  57".73. 
At  the  same  time  on  April  3,  it  was  N.  5  7".  51.  Find  the  mean 
hourly  rate  at  local  apparent  noon  in  Long.  75°  W.  April  2. 


Here  ff1  =      57".  73 


57". 51 


9  -  ff,  =     -0".22 


At  local  apparent  noon  in  longitude  75°  W. 
G.  A.  T.  =  t  (in  formula)  =  +  5h,  and__  =  + 


ffl  + 


x        =  N.  57".  73  +  (—".01)  (2>>.5) 


=  N.  57".  73  —".025  =  N.  57".  705 
Mean  hourly  difference  =  N.  57".  705. 


Ex.  94.  —  On  April  2,  1905,  at  Greenwich  apparent  noon 
the  sun's  H.  D.  of  declination  was  N.  57".73;  at  the  same 
time  April  3,  it  was  N.  57".51.  Find  the  mean  hourly  rate 
and  the  correction  of  the  declination  for  local  apparent  noon 
April  3,  at  a  place  in  longitude  45°  E. 


As  before,^  =  57".  73 
#.,  =  57".  51 


— 0".01 


"At  local  apparent  noon  in  longitude  45°  E. 
G.  A.  T.  =  t  (in  formula)  =  —  3^  and  -  =  —  I*.  5, 


- 


=N.57".51 


=  N.  57". 51  +  ".015  =  N.  57". 525, 
.Mean  hourly  rate =  N.  57". 525. 

The  correction  =  /JT2  +  H*  ~  ^  x -M  t  =  N.  57". 525  x   (— 3h) 

=  S.  172". 575  =  S.  2'  52". 575. 

The  mean  hourly  rate,  when  working  forward,  being 
H!  ±  H*^AHl  X  ty  and  the  correction  being  -j  Hl  ±H*~Hl  X  £  [  t* 

£0.  /&  (  /&4:  fy  ) 

we  see  that  the  expression  for  the  correction  corresponds  to 
that  for  uniformly  accelerated  or  retarded  motion  in 
mechanics, 


V0  representing  the  initial  velocity  or  change  of  the  element; 


366  NAUTICAL  ASTRONOMY 

a,  the  acceleration  or  retardation,  or  the  increase  or  decrease 
of  the  change  per  unit  of  time  next  smaller  than  that  for 
which  V0  is  given;  and  Sf  the  correction. 

This  formula  is  general  in  its  application,  but  it  must  be 
remembered  that,  if,  as  in  the  case  of  the  sun,  V0  is  a  difference 
for  one  hour,  given  in  the  Almanac  for  each  day,  and  taken 
at  the  nearest  Greenwich  date,  a  will  be  -fa  of  the  change  in 
V0  for  24  hours,  in  other  words,  the  hourly  change  of  V0 ;  and, 
if,  as  in  the  case  of  the  moon,  V0  is  the  difference  for  one 
minute  given  in  the  Nautical  Almanac  for  each  hour,  a  will 
be  -g-V  of  the  change  in  V0  for  60  minutes,  or,  in  other  words, 
the  change  in  V0  for  one  minute  of  time. 

In  case  the  given  time  is  nearer  to  a  subsequent  date  than  a 
preceding  Almanac  date,  the  formula  may  be  used,  working 
backwards;  V0  will  be  the  quantity  for  the  subsequent  date  in 
the  Almanac  and  t  the  time  before  this  date. 

Taking  the  first  of  the  two  preceding  examples  to  find  the 
correction, 

8  =  correction  =  j  N.  57".73  +  (—  0".01  X  |)  }  5 

=  K  288".525  =  N.  4'  48".525. 
Taking  the  second  one,  we  have 

8  —  correction  =  J  K  57".51  +  (—  0".01  X  (—  £))  i  (  —  3) 
=  K  57".525  X  (—  3)  =  S.  2'  52".575. 

185.  To  find  from  the  Almanac  a  certain  element  for  a 
given  mean  time  at  a  given  place. 

(1)  Find  the  Greenwich  mean  time  corresponding  to  the 
local  mean  time,  as  previously  explained. 

(2)  Take  from  the  Nautical  Almanac,  for  the  nearest  mean 
time  date  preceding  the  given  Greemvich  mean  time,  the  re- 
quired quantity,  and  the  corresponding   "  difference   for  1 
hour''  or  "  difference  for  1  minute,"  noting  the  name  and  sign 


NAUTICAL  ALMANAC  367 

of  each.  Multiply  the  "difference  for  1  hour"  by  the  hours 
and  decimals  of  an  hour,  or  the  "  difference  for  1  minute  "  by 
the  minutes  and  decimals  of  a  minute  of  the  remaining  Green- 
wich mean  time.  Apply  this  quantity  algebraically  according 
to  sign  to  the  quantity  first  taken  out.  Or,  take  out  the  quan- 
tity for  the  nearest  subsequent  date,  and  the  proper  difference. 
Multiply  this  difference  by  the  fraction  of  time  from  the  given 
Greenwich  date  to  the  Almanac  date,  then  subtract  the  product 
algebraically. 

To  take  out  the  R.  A.  M.  O  or  "  sidereal  time  of  mean 
noon." — Find  what  it  is  for  the  Greenwich  mean  noon  pre- 
ceding the  given  Greenwich  mean  time ;  at  the  bottom  of  the 
column  will  be  found  the  difference  for  1  hour  =  9s.  85  6  5. 
This,  multiplied  by  the  hours  and  decimals  of  an  hour  of  the 
Greenwich  mean  time,  will  give  the  correction  to  be  added  to 
the  quantity  first  taken  out.  Table  III  at  the  end  of  the  Nau- 
tical Almanac  for  converting  a  mean  solar  into  a  sidereal  time 
interval  should  be  used  for  finding  this  correction. 

The  quantities  given  in  the  Ephemeris  for  Washington  mean 
time  may  be  taken  out  in  the  same  way,  first  finding  the 
Washington  mean  time  corresponding  to  the  given  local  mean 
time. 

Ex.  95.— For  a  L.  M.  T.  January  8,  1905,  8h  16m  54s  a.  m., 
in  longitude  80°  31'  30"  W.,  find  the  sun'£  right  ascension  and 
declination,  also  semi-diameter,  equation  of  time,  and  right 
ascension  of  mean  sun,  using  first  differences  only. 

First  find  the  Greenwich  mean  time. 

h        in        s 

Local  astronomical  mean  time  Jan.  1,  20  16  54 

Longitude  from  Greenwich  West  +    5  22  06 

Greenwich  mean  time  Jan.  8,  1  39  00 

Jan.  8,  P.65 

For  mean  time  use  page  II. 

NOTE.—  All  the  data  from  the  Nautical  Almanac  of  1905  necessary  for  the  solution  of 
examples  in  this  book  may  be  found  on  pages  729-738, 


368  NAUTICAL  ASTRONOMY 

To  take  out  the  sun's  right  ascension. 

Sun's  R.  A.  H.  D. 


h       in          a 

Jan.  8,  at  Greenwich  mean  noon         19  16  10.62  10S.925 

H.  D.  108.925  X  lh.65  +  18.03      G.  M.  T.     lh.65 


Required  R.  A.  of  the  sun  19  16  28.65     Corr.         18S.03 

To  find  the  sun's  declination. 

Sun's  Dec.  H.  D. 


o       /          " 

Jan.  8,  at  Greenwich  mean  noon      S  22  17  39.3  N     19".85 

H.  D.  N  19".85  X  lh.65  N  32.75      G.  M.  T.      lh.65 


Required  declination  of  the  sun      S  22  17  06.55      Corr.  N    32".75 

To  find  the  sun's  semi-diameter. 

The  change  of  sun's  semi-diameter  is  so  small,  even  in  24 
hours,  that  it  is  tabulated  only  for  Greenwich  apparent  noon 
on  Page  I.  In  actual  practice  it  would  only  be  taken  out 
to  the  nearest  second  of  arc. 

January  8,  sun's  S.  D.  =  16'  17".75  or  16'  18". 

To  find  the  equation  of  time. 

Eq.  of  time.  H.  D. 


Jan.  8,  at  Greenwich  mean  noon  6  42.54  +  ls.068 

H.  D.  18.068  X  lh.65    '  +        1.76      G.  M.  T.     lh.6b 


Required  equation  of  time  6  44.30      Corr.     + 19.76 

( — )  to  mean  time. 

To  find  the  right  ascension  of  the  mean  sun. 

R.  A.  M.  o  H.  D. 


h  in          a 

Jan.  8,  at  Greenwich  mean  noon          19  09  28.08                      98.8565 
H.  D.    98.8565  X  lh.65    (or  using 

Table  III)  16.26        G.  M.  T.     lh.65 


Required  R.  A.  M.  Q  19  09  44.34       Corr.         168.26 


NAUTICAL  ALMANAC  369 

Ex.  96. — Find  the  right  ascension,  decimation,  semi-diam- 
eter, and  horizontal  parallax  of  the  moon  for  1905,  January 
3,  L.  M.  T.  10h  10m  06s  p.  m.  in  longitude  45°  East;  also 
the  right  ascension  and  declination  of  the  planet  Jupiter. 

h       m       s 

Local  astronomical  mean  time  Jan.  3,  10  10  06 

Longitude  from  Greenwich  East,  ( — )   3  00  00 

Greenwich  astronomical  mean  time  Jan.  3,        7  10  06 

=  7h  10m.l  =  7M7 
Moon's  R.  A.  M.  D. 


Jan.  3,  at  7  hrs.  of  G.  M.  T.  17  12  25.60  2S.3443 

M.  D.   2S.3443  X  10m.l  =  +  23.68  10m.l 


Jan.  3,  at  7h  10m  06s  of  G.  M.  T.    R.  A.  =        17  12  49.28        +  23S.68 

Moon's  Declination.  M.  D. 


o       /          // 

Jan.  3,  at  7  hrs,  of  G.  M.  T.  S  17  56  59.3         S  3".156 

M.  D.  3".156  S  X  10m.l  =  S  31.88  10m.l 


Jan.  3,  at  7h  10m  06s  of  G.  M.  T.   Dec.  =    S  17  57  31.18        S  31".88 

The  moon's  semi-diameter  for  G.  M.  T.  7h  10m  06s  Janu- 
ary 3. — The  moon's  semi-diameter  is  tabulated  only  for  noon 
and  midnight;  therefore,  if  the  given  G.  M.  T.  is  less  than  12 
hours,  take  from  the  noon  column ;  if  G.  M.  T.  is  greater  than 
12  hours,  take  from  the  midnight  column.  In  the  former  case, 
divide  the  difference  between  the  semi-diameters  at  noon,  and 
at  midnight  by  12,  and  multiply  quotient  by  the  hours  and 
decimals  of  G.  M.  T.  for  the  change.  In  the  latter  case,  divide 
the  difference  between  the  semi-diameters  at  midnight  and 
following  noon  by  12,  and  multiply  quotient  by  hours  and 
decimals  in  excess  of  12  hours  for  the  change. 


370  NAUTICAL  ASTRONOMY 

For  moon's  semi-diameter. 

i     //  i     a 

Jan.  3,  at  noon  S.  D.          15  44.4      At  midnight  S.  D.  15  40.6 

Decrease  in  7M7  =  2.3      At  noon  S.  D.  15  44.4 


S.D.  at  given  G.M.T.  =  15  42.1      Decrease  in  12h  =  3.8 

"      lh=  0.32 

«      71*17  —  2.29 

Moon's  horizontal  parallax  for  G.  M.  T.  7h  10m  06s  January  3. 

Jan.  3,  at  noon  H.  P.        57  40        Diff.  for  1  hour  (— )  1".13 

Correction  (— )         8.1      G.  M.  T.  7M7 


H.  P.  at  given  G.  M.  T.    57  31.9     Correction  (— )  8."10 

It  will  be  noticed  that  the  moon's  H.  P.  is  taken  out  for 
noon  or  midnight,  according  as  the  G.  M.  T.  is  less  or  greater 
than  12  hours,  and  corrected  by  the  difference  for  one  hour, 
multiplied  by  the  remaining  hours  and  decimals  of  the  Green- 
wich mean  time. 

To  find  the  right  ascension  of  the  planet  Jupiter. 

Jupiter' sR.  A.  H.  D. 


h      in  s 

Jan.  3,  at  Greenwich  mean  noon  1  18  55.07  +  Os.571 

H.  D.  +  08.571  X  7M7  =  +  4.09     G.  M.  T.     7h.17 


Required  R.  A.  of  Jupiter  1  18  59.16       Corr.         48.094 

To  find  the  declination  of  Jupiter. 

Jupiter's  declination.  H.  D. 


o      /  // 

Jan.  3,  at  Greenwich  mean  noon         N  6  57  41.2  N     4".2G 

H.  D.  N  4".26  X  7M7  =  N  30.5       G.  M.  T.    7M7 


Required  declination  of  Jupiter          N  6  58  11.7       Corr.  N  30".54 


NAUTICAL  ALMANAC 


371 


For  a  given  mean  time  to  find  the  right  ascension  and  decli- 
nation of  a  star. 

Ex.  97. — Let  these  elements  of  the  star  Arcturus  (a  Bootis) 
be  required  for  the  L.  M.  T.,  1905,  January  19,  lh  48ra  15s 
a.  m..,  at  a  place  in  longitude  40°  15'  W. 

A  mean  place  table  (N.  A.  for  1905,  pp.  304-312)  is  used 
as  an  index,  and  shows  on  page  308  for  Arcturus  an  approxi- 
mate E.  A.  14h  llm  19s. 7.  The  apparent  right  ascension  and 
decimation  are  found  on  page  368,  in  a  table,  pp.  324  to  400, 
in  which  fixed  stars  are  tabulated  in  order  of  right  ascension 
for  Washington  mean  time  at  intervals  of  ten  days.  The 
right  ascension  and  declination  of  Arcturus  are  there  found 
for  W.  M.  T.  January  20d.8  as  follows : 

E.  A.  14h  llm  19S.29,  change  for  10  days  +  Os.32. 

Dec.  K  19°.  40'  31".l,  change  for  10  days  S.  2".0. 


To  find  the  Washington  mean  time. 


Local  civil  mean  time  Jan.  19  (a.  m.) 
Local  astronomical  mean  time  Jan.  18 
Longitude  West 

G.  M.  T.  corresponding  Jan.  18 
Longitude  of  Washington  West 

Washington  mean  time  Jan.  18 
=  Jan.  18d.47. 


h  m  8 

1  48  15 

13  48  15 

+    2  41  00 


16  29  15 
)    5  08  15.78 

11  20  59.22 


Diff.  of  Washington 

time. 

Star's  Right  Ascension. 

Tabulated  time,  Jan. 
Given  time,         Jan. 

d 
20.8 

18.47 

h     m       s 
14    11  19.29 

Corr.                  —  .075 

s 
Change  in  ld  —  +  0.032 

Interval          (—  )  2*.  33 

3 

Correction       (—  )  0.075 

Interval,               (—  ) 

2.33 

R.  A.  —  14  11  19.215 

Star's  Declination. 

0        /            // 

N   19  40  31.1 
Corr.  N                  0.5 

it 
Change  in  Id  —  S  0.2 

Interval      (—  )     2d.83 
Correction  N         0.466 

Dec.  N    19  40  81.6 

372  NAUTICAL  ASTRONOMY 

Except  in  the  case  of  Polaris,  it  is  not  usual  to  take  out  the 
E.  A.  and  decimation  of  stars  with  such  precision;  the  ele- 
ments as  tabulated  for  the  nearest  day  being  used  in  observa- 
tions of  the  stars  at  sea.  The  above  method  is  given  for  use 
in  those  cases  where  extreme  precision  might  be  required. 

186.  To  find  from  the  Almanac  a  certain  element  of  the 
Gun  for  a  given  local  apparent  time. — (1)   Find  the  corres- 
ponding Greenwich  apparent  time;  to  do  which  express  the 
local .  apparent  time  astronomically,  applying  to  it  the  longi- 
tude, plus  if  West,  minus  if  East,  as  already  explained  for 
finding  G.  M.  T. 

The  elements  are  then  to  be  taken  from  page  I  of  the  Al- 
manac where  they  are  given  for  apparent  noon. 

187.  To  find  a  certain  element  of  the  sun  when  it  is  on 
the  meridian  of  a  given  place  or  at  local  apparent  noon. — 

Proceed  exactly  as  explained  above. 

The  most  common  use  of  this  problem  is  when  finding  the 
sun's  declination  in  the  case  of  a  meridian  altitude  of  the 
sun,  of  an  altitude  near  noon,  or  in  the  case  of  finding  declina- 
tion of  sun  and  equation  of  time  in  equal  altitudes  for 
chronometer  error. 

At  the  instant  of  apparent  noon,  the  local  apparent  time  is 
Qh  0m  QS>  Therefore,  if  in  longitude  60°  W.  on  January  5,  we 
have  h  m  9 

Local  astronomical  apparent  time  Jan.  5,  0  00  00 

Longitude  West  +  4  00  00 


Greenwich  ast.  apparent  time  Jan.  5,  4  00  00 

But  if  in  longitude  60°  E.  at  local  apparent  noon  on  Janu- 
ary 5,  we  would  have 

h     m       s 

Local  astronomical  apparent  time  Jan.  5,  0  00  00 

Longitude  East  (— )  4  00  00 

Greenwich  apparent  time  Jan.  5,  ( — )  4  00  00 

Or  Jan.  4,  20  00  00 


NAUTICAL  ALMANAC 


373 


From  the  above  it  is  clear  that  in  longitude  West,  the 
G.  A.  T.  of  local  noon  is  equal  to  the  longitude,  or  it  is  after 
noon  of  the  same  date  by  the  number  of  hours  in  the  longi- 
tude; but  that  in  East  longitude  at  local  apparent  noon  the 
G-.  A.  T.  is  before  the  noon  of  local  date  by  the  number  of 
hours  in  the  longitude,  or  G.  A.  T.  =  ( — )  longitude. 

Hence  enter  Page  I  and  take  out,  for  Greenwich  noon  of  the 
same  date  as  the  local  civil  date,  the  required  quantities;  mul- 
tiply the  hourly  difference  by  the  hours  and  decimals  of  longi- 
tude; apply  the  correction  for  a  time  after  noon  if  longitude 
is  West,  for  a  time  before  noon,  if  longitude  is  Bast,  noting 
whether  the  quantities  are  increasing  or  decreasing  for  times 
after  or  before  noon,  and  applying  the  corrections  accordingly. 

Ex.  98. — Find  the  sun's  declination  and  equation  of  time 
for  local  apparent  noon  at  a  place  in  longitude  5h.l  W.  on 
January  2,  1905. 


Times. 


Sun's  declination.       H.  D. 


Eq.  of  T. 


H.  D. 


o     *     ••  it 

At  G.  A.  noon  Jan.  2,      S  22  57  16.3          N    13.13 


Corr.  N 


5.1 


At  L.  A.  noon  Jan.  2,        S  22  56  09.34  Corr.  66".96 


111         fl  8 

4  00.41  +  1.176 

h 
Corr.  +     6.00     A  =  +  5.1 

4  06.41  Corr.+  5.998 
+  to  A  pp.  T. 


374 


NAUTICAL  ASTRONOMY 


Ex.  99.— January  11,  1905,  in  longitude  96°  08'  51"  W., 
find  the  sun's  declination  and  equation  of  time  at  local  appar- 
ent noon,  using  2d  differences. 

Here  longitude  —  6h  24m  35S.4  W.  =  6h.41  W.  =  G-.  A.  T. 
of  local  apparent  noon. 


Times.                    Sun's  Dec. 

H.  D.  at  G.  A.  Noon  and  Change. 

0         •     •  • 

At  G.  A.  noon  Jan.  11,          S  21  51  50 
Corr.  N       2  28.97 
At  L.  A.  noon  Jan.  11,         S  21  49  21.03 

Jan.  11,  H.  D. 
Jan.  12,      " 

Change  in  24  hours 
Change  in  1   hour 

Change  in  -^-hrs.  = 
Jan.  11,  H.  D. 
Mean  H.  D. 
G.  A.  T.  =  A  W 

N    23.10 
>    24.16 

(+)      1.06 
(+)      0.0442 

3h.2  W(+)      0.141 
N    23.10 

N    23.241 
=    6h.41 

Corr.  N  148"  .97 

Times.                     Eq.  of  T. 

H.  D.  at  G.  A.  Noon  and  Change- 

m    s 

At  G.  A.  noon  Jan.  11,                   7  57.14 
Corr.  +      6.38 

Jan.  11,  H.  D. 
Jan.  12,      " 
Change  in  24  hours 
Change  in  ——hrs. 
Jan.  11.  H.  D. 
Mean  H.  D. 
G.  A.  T.  =  A  W  = 

+      0.998 
+      0.972 

At  L.  A.  noon  Jan.  11,                    8  03.52 
+  to  A  pp.  time 

(-)      0.026 
(-)      0.003 
+      0.998 

1  +      0.995 

Corr.  +     6».38 

When  longitude  is  East,  the  Greenwich  apparent  time  of 
local  noon  equals  ( — )  longitude  of  the  local  civil  date. 


NAUTICAL  ALMANAC 


375 


Ex.  100.— April  4,  1905,  in  longitude  10h  04m  498.6  East, 
find  the  sun's  declination  and  equation  of  time  at  local  ap- 
parent noon,  using  2d  differences.  Here  the  G.  A.  T.  of  local 
apparent  noon  equals  ( — )  10h.08. 


Times                      Sun's  Dec. 

H.  D.  at  G.  A.  Noon  and  Change. 

0    /        If 

At  G.  A.  noon  April  4,      N    6  33  35.6 
Corr.  8        9  37.87 

April  3,  H.  D. 
April  4,    " 
Change  in  24  hours 
Change  in  1  hour 

Change  in  -^  hrs. 
April4,H.D. 
Mean  H.  D. 
G.  A.  T.  =  (—  )  X  = 
Correction 

n 
N    57.61 
N    57.28 
(-)      0.23 
(-)      0.0096 

(  +  )      0.048 
N    57.28 

At  L.  A.  noon  April  4,      N   5  22  57.73 

N    57.328 
(-)  IQh  .08 

S  577".87 

Times.                        Bq.  of  T. 

H.  D.  at  -G:  A.  Noon  and 

Change. 

m      B 

At  G.  A.  noon  April  4,                  3  10.82 
Corr.  +       7.43 

April  3,  H.  D. 
April  4,     " 
Change  in  24  hours 
Change  in  |  hrs. 
April  4,  H.  D. 
Mean  H.  D. 
G.  A.  T.  =  (-)  A  = 
Correction 

(—  )    "  0.742 
(—  )      0.736 

At  L.  A.  noon  April  4,                  3  18.25 

(+)     0.006 
(—)      0.001 
(—  )      0.736 

(  —  )    0.737 
(-)10h.08 

+     78.43 

To  find  the  sun's  declination  and  equation  of  time  at  local 
apparent  midnight,  proceed  as  in  the  above  examples,  using 
for  G.  A.  T.  in  the  first  of  the  two  preceding  examples 
(12  hrs.  +  A)  —  18h.41  January  11,  and  in  the  second 
(12  hrs.  —  A)  =  12  hrs.  —  10h.08  =  lh.92  April  4. 

188.  To  find  the  local  mean  time  of  transit  of  the  moon 
over  a  given  meridian  on  a  given  date,  and  the  moon's  right 
ascension,  declination,  semi-diameter,  and  horizontal  parallax 
at  that  instant. — The  Nautical  Almanac,  page  IV  of  each 
month,  contains  the  Greenwich  mean  time  of  each  transit  of 


376  NAUTICAL  ASTRONOMY 

the  moon  over  the  meridian  of  Greenwich.  This  time  is  the 
hour  angle  of  the  mean  sun  when  the  moon  is  on  the  meridian, 
and,  therefore,  equals  the  difference  of  the  right  ascensions  of 
the  moon  and  mean  sun.  Both  the  moon  and  mean  sun  in- 
crease their  right  ascensions  daily,  but  the  increase  for  the 
moon  is  greater  than  that  for  the  mean  sun,  so  that  each  day 
the  moon  gets  further  and  further  to  the  eastward  of  the 
mean  sun,  and  in  the  diurnal  revolution  comes  to  the  meridian 
later  each  day  than  on  the  preceding  day;  the  number  of 
minutes  varying  with  the  moon's  motion,  but  approximating 
an  average  value  of  50  minutes. 

This  retardation,  represented  by  ~R,  occurs  during  a  passage 

of  the  moon  over  24  hours  of  longitude,  and  for  any  longitude 

r> 
X  hrs.  the  retardation  will  be  —  X  A,  so  it  is  easily  seen  that 

if  there  were  no  retardation  whatever,  the  local  time  of  the 
moon's  meridian  passage  in  any  longitude  would  be  the  same 
as  that  at  Greenwich,  but  there  is  retardation,  and  the  local 
mean  time  of  transit  over  a  meridian  is  gotten  from  the 
Greenwich  mean  time  of  Greenwich  transit  by  computing  the 
amount  of  retardation  corresponding  to  the  number  of  hours 
and  decimals  of  an  hour  of  longitude,  and  applying  it  to  the 
Greenwich  mean  time  of  Greenwich  transit,  adding  that 
amount  for  west  longitude,  subtracting  for  east  longitude; 
west  longitude  being  regarded  as  -J-,  east  longitude  as  ( — ) . 

r> 

The  value  of  ~ ,  or  the  hourly  retardation,  is  given  in  its 
Z4. 

appropriate  column  opposite  the  time  of  transit  on  page  IY. 
The  reduction  for  longitude  is  tabulated  in  table  11,  Bowditch, 
the  arguments  being  "  longitude/'  and  "  daily  variation  of 
the  moon's  passing  the  meridian." 

The  times  given  in  the  Almanac  are  for  the  astronomical 
date,  and  care  must  be  exercised  in  finding  the  meridian  pas- 
sage on  a  given  civil  date;  hence  the  rules: 


NAUTICAL  ALMANAC 


377 


(1)  Take  the  time  of  the  moon's  meridian-  passage  from 
the  Nautical  Almanac  for  the  given  civil  date  when  the  time 
tabulated  plus  the  correction  for  retardation  is  less  than  12 
hours,  because  then  the  astronomical  date  is  the  same  as  the 
civil  date. 

(2)  When  it  is  seen  that  the  sum  of  the  tabulated  time  of 
passage  plus  the  correction  for  retardation  on  the  given  civil 
date  will  be  greater  than  12  hours,  take  out  the  time  of  pas- 
sage for  the  day  before,  since  in  this  case  the  astronomical 
date  is  one  day  less  than  the  civil  date. 

(3)  Multiply  the  " Diff.  for  1  hour"  by  the  longitude  in 
hours  and  decimals,  adding  the  product  to  the  G.  M.  T.  of 
meridian  passage  at  Greenwich   when  longitude  is  West,  sub- 
tracting if  longitude  is  East.     The  result  will  be  the  local 
mean  time  of  local  transit  (see  Art.  196  (c) ). 

Ex.  101.— In  longitude  100°  30'  W.  find  the  time  of  me- 
ridian transit  of  the  moon  for  1905,  January  19  (civil  date), 
then  the  corresponding  Gr.  M.  T. 


Meridian  Transit  of  the  Moon. 

Retardation. 

o     r 

Long.  100  30  W 
or 
Long.  6h  42m  w 
=  6h.7  W 

G.  M.  T.  of  Gr.  transit  Jan.  19, 
Corr.  for  longitude  West 
L.  M.  T.  of  local  transit  Jan.  19, 
Longitude  West 
G.  M.  T.  of  local  transit  Jan.  19, 

h     m 
10  53.8 
+      15.88 
11  09.68 
+   6  42 

For  lh,  2m.37 

\  =     +    6h.7 

Corr.+  15m.88 

17  51.68 

Ex.  102. — In  longitude  100°  30'  E.  find  the  time  of  me- 
ridian transit  of  the  moon  for  1905,  January  22  (civil  date), 
then  the  corresponding  G.  M.  T. 


Meridian  Transit  of  the  Moon. 


Long.  100°  30'  E          G.  M.  T  of  Gr.  transit  Jan.  21,         12»»  49m 
or  Corr.  for  longitude  East  (-)       16.01 

Long.  6h  42m  E  L.  M.  T.  of  local  transit  Jan.  21,       12    32.99 

=  6h.7  E  Longitude  East  (— )    6    42 

G.  M.  T.  of  local  transit  Jan.  21,        5    50.99 


Retardation. 


For    lh,2m.39 


Corr.— 16m  .01 


378  XAUTICAL  ASTRONOMY 

The  times  of  transits  at  Greenwich  are  given  only  to  the 
nearest  tenth  of  a  minute,  and  the  resulting  local  time  of  local 
transit  will  be  only  approximate,  though  sufficiently  exact  for 
navigators.  A  more  exact  time  may  be  found  by  first  finding 
the  approximate  L.  M.  T.  of  local  transit  and  then  the  approxi- 
mate G.  M.  T.  of  local  transit  for  which  the  moon's  right 
ascension  may  be  taken  out. 

This  right  ascension  is  the  local  sidereal  time  of  the  moon's 
local  transit  and  the  local  mean  time  corresponding  may  be 
found  (see  Ex.  132). 

If  then  other  elements  are  desired  at  the  time  of  the  moon's 
local  transit,  find  the  G.  M.  T.  corresponding  to  the  L.  M.  T. 
just  found,  and  take  out  for  this  G.  M.  T.  the  required  ele- 
ments. 

189.  To  find  the  local  mean  time  of  transit  of  a  planet 
over  a  given  meridian  on  a  given  date,  and  the  correspond- 
ing G.  M.  T.,  also  the  planet's  right  ascension  and  declination 
at  that  instant. — The  mean  time  of  each  meridian  transit  for 
the  meridian  of  Greenwich  is  given  in  the  Almanac  for  each 
of  the  seven  major  planets.  On  certain  dates  there  may  be 
retardation,  on  others  acceleration,  in  the  times  of  return  to 
the  meridian.  In  the  case  of  a  retardation,  the  time  of  local 
transit  is  found  as  in  the  case  of  the  moon;  in  the  case  of 
acceleration,  the  sign  of  the  reduction  for  longitude  is  re- 
versed. Or,  considering  the  hourly  retardation  +,  the  hourly 
acceleration  ( — ) ,  west  longitude  +,  and  east  longitude  ( — ) , 
the  rule  of  signs  will  determine  the  sign  of  the  reduction. 

Having  found  the  L.  M.  T.  of  local  transit,  deduce  the 
G.  M.  T.  by  applying  the  longitude,  and  take  out  for  this 
G.  M.  T.  the  planet's  right  ascension  and  declination  (Art. 
185,  Ex.  96). 

Having  found  the  right  ascension  of  a  planet  when  it  is  on 
the  meridian,  take  this  as  local  sidereal  time  and  find  the 
corresponding  local  mean  time ;  the  result  will  be  closer  than 


NAUTICAL  ALMANAC 


379 


the  time  tabulated  which,  however,  is  sufficiently  exact  for 
navigators. 

It  will  be  noticed  that,  while  in  the  case  of  the  moon  the 
retardation  was  given  for  one  hour,  in  the  cases  of  planets, 
the  retardation  or  acceleration  will  be  obtained  for  24  hours 
by  taking  the  difference  of  the  times  of  Greenwich  transit  for 
the  given  day  and  the  day  following  when  in  West  longitude, 
the  difference  of  those  times  for  the  given  day  and  day  preced- 
ing when  in  East  longitude. 

The  change  for  1  hour  will  then  be  one  twenty-fourth  of 
the  difference  for  24  hours.  This  hourly  change  multiplied 
by  the  hours  and  decimals  of  an  hour  in  the  longitude  will 
be  the  change  for  longitude. 

Ex.  103.— In  longitude  75°  W.  find  the  L.  M.  T.  of  transit 
of  Jupiter,  1905,  January  4,  civil  date;  also  right  ascension 
and  declination  of  Jupiter  for  that  instant. 


Meridian  Transit  of  Jupiter. 

Difference. 

Long.  75°  W 

=  5''  W.  At  Gr.  noon  Jan.  4, 
Correction  for  longitude  W. 
L.  M.  T.  of  local  transit  Jan. 
Longitude  West 
G.  M.  T.  of  local  transit  Jan. 

h   m 
6  24.5 

-    0.77 

h  m 
for  24  =  —  3.7 

for  5  =  —0.77 

4,    6  23.73 

+  5 

4,  11  23.73 

Times. 

Jupiter's 
R.A. 

H.  D. 

Jupiter's  Dec. 

H.  D. 

At  Gr.  noon  Jan.  4. 
Corr.  for  G.  M.  T. 
At  local  transit 

h  m    s 
1  19  09.14 

+      6.85 

+            Os.601 
G.M.T.    11M 

0      1      It 

N  6  59  25.5 
Corr.  N          50.5 

N  4".43 
G.M.T.  11M 

1  19  15.99 

Corr.  +  68.85 

N  700  16 

Corr.  N  60".50 

CHAPTER  XIV. 

RELATION  OF  MEAN,  APPARENT,  AND  SIDEREAL 
TIMES.—  CONVERSION  OF  TIME.—  RELATION  OF 
TIME,  HOUR  ANGLES,  AND  RIGHT  ASCENSIONS, 
AND  A  CONSIDERATION  OF  PROBLEMS  INVOLVING 
THEM.—  FINDING  LOCAL  AND  WATCH  TIMES  OF  A 
BODY'S  TRANSIT,  ETC. 

190.  To  interconvert  apparent  and  mean  time.  —  The  equa- 
tion of  time  being  the  difference  between  the  hour  angles  of 
the  true  and  mean  suns,  or,  in  other  words,  between  apparent 
and  mean  times,  when  one  is  given,  the  other  is  obtained  by 
applying  the  equation  of  time  with  its  proper  sign  of  appli- 
cation to  the  given  time.  Thus,  if  for  the  same  instant, 

tm  represents  local  mean  time, 

ta  represents  local  apparent  time, 

E  represents  equation  of  time  with  positive  sign  of  applica- 
tion to  apparent  time, 


Hence  for  the  given  local  time  (apparent  or  mean),  ex- 
pressed astronomically,  find  the  Greenwich  time  (apparent  or 
mean).  Take  out  of  the  Nautical  Almanac  for  the  Green- 
wich instant  the  equation  of  time  (from  page  I  of  the  required 
month  when  apparent  time  is  given,  or  page  II  when  mean 
time  is  given).  The  reduction  then  is  made  by  applying  the 
corrected  equation  of  time  to  the  given  time,  with  the  proper 
sign  as  shown  at  the  top  of  the  column  in  which  it  is  found, 


MEAN  AND  APPARENT  TIMES  381 

The  equation  of  time  found  on  page  I,  Nautical  Almanac, 
is  the  mean  time  of  apparent  noon  at  Greenwich,  and,  if  cor- 
rected for  longitude,  it  is  the  mean  time  of  local  apparent 
noon. 

The  equation  of  time  found  on  page  II,  Nautical  Almanac, 
is  the  apparent  time  of  mean  noon  at  Greenwich,  and  if  cor- 
rected for  longitude  it  is  the  apparent  time  of  local  mean  noon. 

Ex.  104.—  January  2,  1905,  in  longitude  75°  30'  W.,  find 
the  local  apparent  time  corresponding  to  a  local  mean  time 
8h  10m  10s  p.  m. 

hms  m    s  h  m  s 

L.  M.  T.    8    10  10  Jan.  2.  Eq.t.atOh=4    0.33  H  D  +K176      L-M.T.=  8  10  10 

Long.        5      2  00  W  Corr.        =  +  15.62  G.  M.  T.  W>.2    Eq.  t.  —        4  15.85 

G.M.T.  13    12  10  Jan.  2.  Eq.  t.       =4  15.85  Corr.  +  15>*.52  L.A.  T.    8  05  54.15 
=  13h.3                                      (—  )  to  M.  T. 

Ex.  105.—  April  3,  1905,  in  Long.  100°  45'  E.,  find  the  local 
mean  time  corresponding  to  5h  10m  a.  m.,  local  apparent  time. 

hms  ms  hms 

L.A.T.  17  10  00  April  2.  Eq.  t.  at  Oh  =  3  46.46  H.D  —  0".748      L.A.T.    17  10  00 

Long.      6  43  00  E  Corr.  =  —  7.82  G.A.T.10M5   Eq.  t  .  3  38.64 

G.A.T.  10  27  00  April  2.  Eq.  t.  =3  38.64  Corr.-7».82  L.M.T.    =  17  13  38.64 

+  to  App.  T.  or  April  3,  a.  m.,    5  13  38.64 

191.  Formulae  for  the  intercon  version  of  mean  and  side- 
real time  intervals.  —  Since  a  sidereal  year  contains  365.25636 
mean  solar  days,  or  366.25636  sidereal  days,  each  unit  of 


v\ 

mean  solar  time  will  contain  o^'f^.o^  sidereal  units  of  the 

o65.  25636 

same  denomination,  or  each  unit  of  sidereal  time  will  contain 
units  °f  mean  time  of  the  same  denomination. 


Since  both  are  uniform  measures  of  time,  any  interval  of 
time  expressed  either  in  mean  solar  or  sidereal  units  may  be 
expressed  in  units  of  the  other  denomination. 

Thus,  if  any  interval  of  time  be  represented  by  t  if  ex- 
pressed in  mean  solar  time,  by  s  if  expressed  in  sidereal  time, 
s  366.25636  _ 


whence  s  =  t  +  .0027379/,  (137) 

t  —  s  —  .00273045,  (138) 


382  NAUTICAL  ASTRONOMY 

and  by  these  formulae  any  interval  of  the  one  kind  of  time 
can  be  converted  into  an  interval  of  the  other  kind  of  time. 

The  reduction  is  facilitated  by  the  use  of  Table  II  of  the 
Nautical  Almanac  for  converting  sidereal  intervals  into  mean 
solar  time  intervals,  which  contains  for  each  second  of  s  the 
value  .0027304^  expressed  in  minutes  and  seconds;  also  by 
Table  III,  for  converting  a  mean  solar  time  interval  into  a 
sidereal  time  interval,  which  contains  for  each  second  of  t  the 
value  .002 73 79 £  expressed  in  minutes  and  seconds.  Tables 
8  and  9  of  Bowditch  are  for  the  same  purpose. 

If  t  and  s  are  in  units  of  hours,  the  above  formulae  become 
s  =  t  (1  +  9S.8565)  =  t  +'  9-.8565$,         (139) 
t  =  s  (1  —  9S.8296)  —  s  —  9s.8296s,         (140) 
so  that  in  the  absence  of  the  above  mentioned  Tables  the  re- 
duction may  still  be  conveniently  calculated. 

Acceleration  and  retardation. — If  in  (137)  t  =  24  hrs., 
s  will  equal  24h  3m  56S.5553;  or  in  a  mean  solar  day  sidereal 
time  gains  on  mean  time  3m  56S.5553,  and  this  is  called  the 
acceleration  of  sidereal  on  mean  time.  If  in  (138)  s  =  24 
hrs.,  t  =  24h  minus  3m  55S.9094,  or  in  a  sidereal  day  mean 
time  loses  on  sidereal  time  3m  55S.9094,  and  this  is  the  retarda- 
tion of  mean  solar  on  sidereal  time. 

Examples  on  the  conversion  of  a  mean  solar  time  interval 
into  a  sidereal  time  interval. 

Ex.  106. — Express  10  hours  of  mean  solar  time  in  sidereal 
time. 

Taking  formula  sh  =  th(l  +  .0027379),  we  have 

s  —  10h.027379  =  10h  Olm  383.564. 

Taking  formula  sh  =  ^h(l  +  9S.8565), 
we  have  s  —  10h  Olm  388.565. 

Using  table  III,  Nautical  Almanac,  we  have 
t  =  a  mean  solar  time  interval,  10h  00m  00s 

From  table  III,  reduction  to  a  sidereal  interval  -j-    1    38  .565 


The  required  sidereal  time  interval  10h  Olm  383.565 


MEAN  AND  AITAUKNT  TIMES  383 

Ex.   101. — Express  15b  33m  29s  of  mean  time  in  sidereal 
time. 

t  =.  a  mean  solar  time  interval  15h  33m  29s 

From  table  III,  reduction  to  a  sidereal  interval  -\- 2     33  .347 

The  required  sidereal  time  interval  15h  36m  02S.347 

Express  in  sidereal  time : 
Ex.  108.—  7h  29m  30S.5  of  mean  time. 

Ans.  7h  30m  448.342  sidereal  time. 
Ex.  109.—  lh  14m  03s  of  mean  time. 

Ans.  lh  14m  15s.  164  sidereal  time. 
Ex.  110.— 23h  15m  10s  of  mean  time. 

Ans.  23h  18m  59M90  sidereal  time. 

Examples  on  the  conversion  of  a  sidereal  interval  into  a 
mean  solar  time  interval. 

Ex.  111. — Express  10h  30m  00s  of  sidereal  time  in  mean 
solar  time. 
Taking  formula  th  =  sh(l  —  .0027304),  we  have 

t  =  ioh.5  —  Oh.0286692  —  10h.471331  =  10h  28m  163.79. 
Taking  formula  th  =  s*(l  —  9S.8296),  we  have 

t  =  10h.5  (1  —  98.8296)  =  10h  30m  —  lm  439.2108 

=  10h  28m  163.789. 

Using  table  II,  Nautical  Almanac,  we  have 
s  =  a  sidereal  time  interval  =  10h  30m  00s 

From  table  II,  reduction  to  a  mean  time  interval  —  1    43  .210 
The  required  mean  solar  time  interval  10h  28m  16S.790 

Express  in  mean  solar  time : 
Ex.  112.— II*  04m  12S.94  of  sidereal  time. 

Ans.  llh  02m  24M25  mean  time. 

Ex.  113. — 15h  08m  33S.37  of  sidereal  time. 

Ans.  15h  06m  043.525  mean  time. 

Ex.  114. — 19h  13m  36S.65  of  sidereal  time. 

Ans.  19h  10m  27'.659  mean  time. 


384  NAUTICAL  ASTRONOMY 

192.  Having  the  mean  time  at  any  place,  to  find  the  cor- 
responding sidereal  time. 

Let  A  represent  the  longitude  of  the  place  expressed  in 

time,  +  when  West,  ( — )  when  East. 
t  the  hour  angle  of  the  mean  sun  expressed  positively 

and,  therefore,  the  local  mean  time. 
(£-|-A)  the  G.  M.  T.  or  elapsed  mean  time  interval 

since  Greenwich  mean  noon. 
8  the  hour  angle  of   T   and  hence  the  local  sidereal 

time. 

(8  +  A)  the  Greenwich  hour  angle  of  T   or  Green- 
wich sidereal  time. 
a0  the  right  ascension  of  the  mean  sun  (E.  A.  M.  O ) 

at  Greenwich  mean  noon. 

If  a  mean  time  interval  since  Greenwich  mean  noon  is 
(^-|-.A)h,  the  corresponding  sidereal  time  interval  will  be 
(£  +  A)h  (1  +  .0037379).  Having  now  the  sidereal  in- 
terval since  Greenwich  mean  noon  and  the  sidereal  time  of 
Greenwich  mean  noon,  or  a-0,  the  Greenwich  sidereal  time 
will  be 

(8  +  A)h  =  fl0  +  (t  +  *)h  (1  +  .0027379) 
8  +  \     =  a0+.\  +  t  +  (t  +  \)  (.0027379) 

S.  T.   }  =  ^  +  t  +  (t  +  X)  (.0027379)       (141) 

The  right-hand  column  of  page  II  of  the  Almanac  contains 
a0  for  each  Greenwich  mean  noon  under  the  head  "  Sidereal 
Time"  or  "Eight  Ascension  of  Mean  Sun."  As  t  +  A  is 
the  G.  M.  T.,  a0  should  be  taken  out  for  Greenwich  mean  noon 
of  the  given  Greenwich  date,  and  corrected  for  the  hours,  min- 
utes, and  seconds  of  Greenwich  mean  time,  using  Table  III  of 
the  Almanac. 

Hence  the  rule :  Express  the  local  mean  time  astronomically 
and  find  the  G.  M.  T.  and  date.  Then  to  the  local  astronomi- 
cal mean  time  add  th&  sidereal  time  or  the  right  ascension  of 


CONVERSION  OF  TIME  385 

the  mean  sun  taken  from  the  Nautical  Almanac  for  noon  of 
the  Greenwich  date,  and  also  the  reduction  from  Table  III 
for  the  hours,  minutes,  and  seconds  of  the  Greenwich  mean 
time.  The  sum,  if  less  than  24  hours,  will  ~be  the  local  sidereal 
time  (L.  8.T.).  If  the  sum  is  greater  than  24  hours,  reject 
24  hours  and  the  remainder  will  be  the  L.  8.  T. 

Since  the  sidereal  time  (R.  A.  M.  O  )  at  Greenwich  mean 
noon,  corrected  for  the  G.  M.  T.  corresponding  to  the  given 
L.  M.  T.,  is  the  right  ascension  of  the  mean  sun  at  the  instant 
of  the  given  L.  M.  T.,  the  above  equation  (141)  is  simply  an 
algebraic  expression  of  what  has  already  been  proven,  namely. 
"  The  sidereal  time  at  a  given  place  is  equal  to  the  right  ascen- 
sion of  the  mean  sun  plus  the  local  mean  time"  (Art.  173). 

It  is  usual  to  keep  the  solar  day;  but  should  it  be  desired 
to  state  the  sidereal  day,  prefix  to  a0  the  sidereal  day  at  the 
instant  of  Greenwich  mean  noon,  which  is  the  same  as  the 
astronomical  day  for  six  months  after  the  vernal  equinox,  one 
day  less  for  six  months  before  the  vernal  equinox.  At  the 
instant  of  the  vernal  equinox,,  the  sidereal  time  and  mean  solar 
time  coincide.  Before  that  time  the  mean  sun  transits  before 
the  vernal  equinox ;  after  that  time,  it  transits  after  the  vernal 
equinox. 

Examples  on  the  conversion  of  local  mean  time  into  local 
sidereal  time. 

Ex.  115.— January  18,  1905,  in  longitude  55°  15'  W.,  the 
local  mean  time  is  8h  06m  29S.5  p.  m.  Find  the  local  sidereal 
time  (see  rule  in  this  Article). 

h     m          8 

The  local  astronomical  mean  time  Jan.  18,  8  06  29.5 

Longitude  from  Greenwich  West  +  3  41  00 

The  Greenwich  mean  time  Jan.  18,  11  47  29.5 

h       m          s 

R.  A.  M.  O  Jan-  18,  at  Greenwich  mean  noon  19  48  53.64 

Reduction  for  G.  M.  T.,  Table  III,  or  98.8565  X  llh.7915  1  56.223 
Add  the  local  astronomical  mean  time  8  06  29.5 


The  required  local  sidereal  time  (rejecting  24  hrs.)        3  57  19.363 


386  NAUTICAL  ASTRONOMY 

Ex.  116. — January  10,  1905,  in  Long.   137°   35'  E.,  the 
L.  M.  T.  is  5h  17m  30s  a.  m.     Find  the  L.  S.  T. 

h       m       s 

Local  astronomical  mean  time  Jan.  9,    '  17  17  30 

Longitude  from  Greenwich  East  9  10  20 


Greenwich  mean  time  Jan.  9,  8  07  10 

h       m  a 

R.  A.  M.  Q  at  Greenwich  mean  noon  Jan.  9,  19  13  24.64 

Reduction  for  G.  M.  T.  Table  III  1  20.029 

Add  the  local  astronomical  mean  time  17  17  30 


Required  local  sidereal  time  (rejecting  24  hrs.j  12  32  14.669 

Ex.  111. — April  16,   1905,  the  Greenwich  mean  time  is 
9h  10m  30s  a.  m.     Find  the  Greenwich  sidereal  time. 

Greenwich  astronomical  mean  time  April  15,  21  10  30 

R.  A.  M.  O  at  Greenwich  mean  noon  April  15,  1  31  53.76 

Reduction  for  G.  M,  T.  Table  III  3  28.711 


Required  Greenwich  sidereal  time  22  45  52.471 

Ex.  118.— January  20,  1905,  at  the  U.  S.  Naval  Academy, 
when  the  75th  meridian  mean  noon  signal  was  received,  a 
sidereal  clock  read  20h  23m  19S.5.  Shortly  after  the  receipt 
of  this  signal  a  comparison  of  this  clock  with  a  mean  time 
chronometer  was:  sidereal  clock,  20h  43m  29s;  mean  time 
chronometer,  5h  24m  16s.  Find  the  error  of  chronometer 
on  G.  M.  T.  (see  example  125,  Art.  193).  Note  that  here  the 
error  of  the  sidereal  clock  is  not  given. 

h       m          s 

At  75th  meridian  mean  noon  sidereal  clock  reads          20  23  19.5 
At  time  of  comparison  sidereal  clock  reads  20  43  29 


Sidereal  interval  since  75th  mer.  mean  noon  0  20  09.5 

Reduction  to  a  mean  time  interval  Table  II  —  3.303 


Mean  time  interval  since  75th  mer.  mean  noon  0  20  06.197 

Longitude  of  75th  meridian  West  +  5  00  00 


Greenwich  mean  time  of  comparison  5  20  06.197 

Chronometer  time  of  comparison  5  24  16 


Error  of  mean  time  chronometer,  fast  on  G.  M.  T.  4  09.80? 


CONVERSION  OF  TIME  387 

Ex.  119.— At  Cebu  I.  Plaza,  Lat.  10°  17'  30"  K,  Long. 
123°  54'  18"  E.,  April  10,  1905,  L.  M.  T.  =  5h  45m  30s  a.  m. 
Find  L.  S.  T. 

h       m       • 

Local  astronomical  mean  time  April  9,  17  45  30 

Longitude  East  from  Greenwich  —  8  15  37.2 


Greenwich  mean  time  April  9,  9  29  52.8 

b     m          • 

R.  A.  M.  Q  at  Greenwich  mean  noon  April  9,  1  08  14.45 

Reduction  for  G.  M.  T.  1  33.616 

Local  astronomical  mean  time  17  45  30 


Local  sidereal  time  18  55  18.066 

Ex.  120.— April  1,  1905,  in  longitude  2h  13m  20s  West,  the 
L.  M.  T.  is  9h  48m  06s  p.  m.  Find  first  the  G.  S.  T.,  then  the 
L.  S.  T. 

h     m       g 

April  1,  the  local  astronomical  mean  time  is  9  48  06 

Longitude  from  Greenwich  West  +  2  13  20 


April  1,  Greenwich  mean  time  12  01  26 

R.  A.  M.  Q  April  1  -at  Greenwich  mean  noon  0  36  42.03 

Reduction  for  G.  M.  T.  Table  III  1  58.513 


Greenwich  sidereal  time  12  40  06.543 

Longitude  from  Greenwich  West  (— )  2  13  20 


Local  sidereal  time  10  26  46.543 

Ex.  121.— January  25, 1905,  at  the  Naval  Academy,  Annap- 
olis, Md.,  in  longitude  5h  05m  56S.5  W.,  when  the  time  signal 
was  received  from  Washington  indicating jnoon  of  75th  merid- 
ian West  longitude,  mean  time,  a  sidereal  clock  read  20h  15m 


388  NAUTICAL  ASTRONOMY 

099.     Kequired  the  error  of  the  sidereal  clock  on  local  sidereal 
time. 

h     m      a 

75th  meridian  mean  time  at  75th  mer.  mean  noon  0  00  00 

Longitude  of  75th  meridian  West  -f  5  00  00 


Jan.  25,  G.  M.  T.  of  75th  mer.  mean  noon  5  00  00 

Longitude  of  Naval  Academy  West  (— )  5  05  56.5 


L.  M.  T.  at  instant  of  75th  mer.  M.  N.  23  54  03.5 

h       m          a 

Jan.  25,  R.  A.  M.  Q  at  G.  M.  noon  20  16  29.54 

Reduction  for  G.  M.  T.  of  75th  mer.  mean  noon  (5  hrs.)        0  49.282 
Local  ast.  mean  time  at  Naval  Academy  23  54  03.5 


Local  sidereal  time  at  Naval  Academy  20  11  22.322 

Reading  of  sidereal  clock  at  the  instant  20  15  09 


Error  of  sidereal  clock  on  L.  S.  T.,  fast,  3  46.678 

193.  Having  the  sidereal  time  at  any  place,  to  find  the 
local  mean  time. — Since  A,  the  longitude,  is  the  G.  M.  T.  of 
local  mean  noon,  or  of  the  instant  when  the  mean  sun  is  on  the 
upper  branch  of  the  local  meridian,  according  to  the  notation 
of  Art.  192, 

a0  +  .0027379A,  will  he  the  local  sidereal  tjme  at  local  mean 

noon. 
8  —  (a0  -f-  .0027379A)   will  be  the  sidereal  interval  since 

noon  as  it  is  L.  S.  T.  —  the  sidereal  time  of  local  mean 

noon. 
[8  —  (a0  +  .0027379A)]  [1  —  .0027304]  will  be  the  mean 

time  interval  since  noon,  and  to  find  L.  M.  T.  it  is  only 

necessary  to  add  the  astronomical  day  to  this  mean,  time 

interval. 

Hence  the  rules: 

(1)  Take  from  ihe  Nautical  Almanac  for  Greenwich  mean 
noon  of  the  given  local  astronomical  day  the  right  ascension  of 


CONVERSION  OF  TIME  389 

the  mean  sun;  apply  to  this  the  reduction  for  longitude  (which 
is  the  change  in  the  mean  sun's  right  ascension  for  that  num- 
ber of  hours)  taken  from  Table  III,  Nautical  Almanac;  add- 
ing for  West,  subtracting  for  East  longitude.  The  result  will 
be  the  right  ascension  of  the  mean  sun  at  local  mean  noon,  or 
sidereal  time  at  that  instant  (local  0  hrs.  of  mean  time). 

(2)  Subtract  this  from  the  given  L.  S.  T.  (adding  £4  hrs. 
to  the  L.  S.  T.  if  necessary  for  subtraction)  and  the  result 
will  be  the  sidereal  interval  from  local  mean  noon. 

(3)  Apply  to  this  the  reduction  of  a  sidereal  to  a  mean 
time  interval  taken  from  Table  11 ' ,  Nautical  Almanac,  which 
is  always  subtractive.     The  result,  after  prefixing  the  given 
astronomical  dayf  is  the  required  local  mean  time. 

In  the  absence  of  Tables,  the  reduction  may  be  made  by 
using  the  formulas  (139)  and  (140)  of  Art.  191. 

Caution. — It  is  much  better  to  convert  a  given  L.  S.  T.  and 
afterwards,  if  desired,  find  the  G.  M.  T.,  than  to  'first  find 
G.  S.  T.  and  then  convert  it  into  G.  M.  T.,  for  the  reason  that 
the  right  ascension  of  the  mean  sun  must  be  taken  out  for  the 
given  astronomical  day.  To  convert  G.  S.  T.  the  Greenwich 
astronomical  date  must  be  known,  and  as  this  may  or  may 
not  be  the  same  as  the  local  astronomical  date,  an  error  might 
result. 

As  a  little  thought  can  easily  determine  the  Greenwich 
date,  this  caution  may  seem  unnecessary  to  those  thoroughly 
familiar  with  the  subject;  to  others,  however,  it  is  most  im- 
portant. 

In  cases  where  the  G.  M.  T.  is  known  in  addition  to  the 
L.  S .  T.,  the  method  of  reduction  is  very  simple. 

From  formula  (141), 

t  =  S  —  K  +  (t  +  A)   (.0027379)]  (142) 


390  NAUTICAL  ASTRONOMY 

Rule:  For  the  G.M.T.  (t  -f  A)  take  out  the  right  ascenr 
sion  of  the  mean  sun.,  subtract  it  from  the  given  L.  8.  T.,  and 
the  result  will  be  L.  M.  T.  of  the  given  astronomical  date. 

So  also  from  Art.  173,  it  is  plain  that  local  apparent  time 
equals  L.  S.  T.  minus  the  apparent  right  ascension  after  cor- 
rection for  Gr.  M.  T.,  if  taken  from  page  II,  or  for  G.  A.  T. 
if  taken  from  page  I  of  the  Nautical  Almanac. 

Examples  on  the  conversion  of  time ;  L.  S.  T.  into  L.  M.  T. 

Ex.  122. — January  8,  a.  m.,  1905,  at  Eoyal  Observatory, 
Lisbon  (Long.  Oh  36m  44S.68  W.),  local  sidereal  time  is  10h 
44m  30s.  Find  the  local  mean  time. 

First  find  the  astronomical  day,  which  is  January  7. 

b      m         9 

R.  A.  M.  Q  or  sidereal  time  at  Greenwich  mean  noon 

Jan.  7,  19  05  31.52 

Reduction  for  longitude  West  +  6.036 


The  sidereal  time  of  local  mean  noon  19  05  37.556 

The   given   local   sidereal   time    ( +  24   hrs.    for   the 

subtraction)  10  44  30 


The  sidereal  interval  from  noon  15  38  52.444 

Reduction  of  a  sid.  to  a  M.  T.  interval  Table  II         —         2  33.812 


The  required  astronomical  mean  time  Jan  7,  15  36  18.632 

Or  civil  time  Jan.  8,  (a.  m.)  3  36  18.632 

Ex.  123.— April  15  (civil  date),  1905,  in  Long.  129°  30' 
45"  E.,  the  local  sidereal  time  is  23h  56m  30s.  Find  the  local 
mean  time. 

The  above  example  does  not  say  whether  it  is  a.  m.  or  p.  m., 
but  the  astronomical  date  must  be  known  before  taking  out 
the  E.  A.  M.  O .  To  determine  this  look  up  the  approximate 
R.  A.  M.  Q,  which  is  found  to  be  about  1J  hours.  Subtract- 
ing this  from  the  L.  S.  T.  leaves  an  approximate  astronomical 


CONVERSION  OF  TIME 


391 


mean  time  of  over  22  hours ;  the  civil  time  is,  therefore,  a.  m., 
and  hence  the  local  astronomical  date  is  April  14. 

ii     in         a 

R.  A.  M.  O  or  sidereal  time  at  Greenwich  mean  noon 

April  14,  1  27  57.21 

Reduction  for  Long.  (8h  38m  03s  East)  Table  III      —        1  25.102 

The  sidereal  time  of  local  mean  noon  1  26  32.108 

The  given  local  sidereal  time  23  56  30 

The  sidereal  interval  from  noon  22  29  57.892 

Reduction  of  a  sid.  to  a  M.  T.  interval  Table  II          —         3  41.159 

The  required  astronomical  L.  M.  T.  April  14,  22  26  16.733 

Or  civil  date  April  15,  (a.  m.)  10  26  16.733 

Ex.  121f-. — On  January  11,  astronomical  time,  1905,  the 
sidereal  clock  time  of  transit  of  a  Leonis  (Regulus)  over  the 
middle  wire  of  a  transit  instrument  at  the  U.  S.  Naval  Acad- 
emy was  10h  05m  2 2s. 5.  Later  a  comparison  of  the  sidereal 
clock  and  a  mean  time  chronometer  was :  Sid.  clock,  10h  30m 
20S.5 ;  M.  T.  chro.,  8h  05m  10s.  Find  the  error  of  chronometer 
on  Gk  M.  T.  Longitude  of  Naval  Academy,  5h  05ra  568.5  West. 


R.  A.  of  >f<  a  Leonis  at  transit  equals  the  L.  S.  T. 
Reading  of  sidereal  clock  at  star's  transit 

Error  of  sidereal  clock  on  L.  S.  T.  fast 
Reading  of  sidereal  clock  at  comparison 

L.  S.  T.  at  instant  of  comparison 

R.  A.  M.  Q  or  sidereal  time  at  G.  M.  noon  Jan.  11, 
Reduction  for  longitude  West  (5h  05m  56S.5) 

Sidereal  time  of  local  0  hrs. 
The  given  L.  S.  T.  at  comparison 

Sidereal  interval  from  noon 

Reduction  of  a  sid.  to  a  M.  T.  interval  Table  II 

The  local  mean  time  at  instant  of  comparison 
Longitude  of  Naval  Academy  West 

G.  M.  T.  at  instant  of  comparison 
Reading  of  M.  T.  chronometer  at  comparison 
Error  of  chronometer  (dropping  12  hrs.),  slow  on 
G.  M.  T. 


h       m  s 

10  03  19.54 

10  05  22.5 

2  02.96 

10  30  20.5 

10  28  17.54 

h      m  g 

19  21  17.75 
50.258 

19  22  08.008 

10  28  17.54 

15  06  09.532 

2  28.452 


15  03  41.08 
+    5  05  56.5 

20  09  37.58 
8  05  10 

4  27.58 


392  NAUTICAL  ASTRONOMY 

The  following  example  worked  under  Art.  192  without  first 
finding  the  error  of  the  sidereal  clock,  by  considering  only  the 
sidereal  interval  from  noon  to  time  of  comparison,  and  finding 
the  corresponding  mean  time  interval  from  noon  and  then 
the  Gr.  M.  T.,  will  now  be  worked  by  finding  the  clock  error  on 
L.  S.  T.  as  indicated  in  the  solution. 

Ex.  125.— January  20,  1905,  at  IT.  S.  Naval  Academy, 
when  the  75th  meridian  mean  noon  signal  was  received,  a 
sidereal  clock  read  20h  23m  19S.5.  Shortly  after  the  receipt 
of  this  signal  a  comparison  of  this  clock  with  a  mean  time 
chronometer  was:  Sid.  clock,  20h  43m  29s;  M.  T.  chro.,  5h 
24m  16s.  Find  the  error  of  the  clock  on  L.  S.  T.  and  the  error 
of  the  chronometer  on  G-.  M.  T.  (see  example  118,  Art.  192). 

h     m       s 

The  75th  mer.  mean  time  of  75th  mer.  mean  noon          0  00  00 

Longitude  of  75th  meridian  West  +    5  00  00 
G.  M.  T.  of  75th  mer.  mean  noon  Jan.  20,  5  00  00 

R.  A..M.  0  or  sidereal  time  G.  M.  noon  Jan.  20,  19  56  46.76 

Correction  for  G.  M.  T.  49.282 


G.  S.  T.  of  75th  meridian  mean  noon  0  57  36.042 

Longitude  of  Naval  Academy  West  5  05  56.5 


L.  S.  T.  of  75th  meridian  mean  noon  19  51  39.542 

Sidereal  clock  time  of  75th  meridian  noon  20  23  19.5 


Error  of  sidereal  clock  on  L.  S.  T.  fast  0  31  39.958 

Sidereal  clock  time  of  comparison  20  43  29 


L.  S.  T.  at  instant  of  comparison  20  11  49.042 

h  m          a 

R.  A.  M.  O  or  sidereal  time  at  G.  M.  noon  Jan.  20,        19  56  46.76 
Reduction  for  longitude  (5h  05m  568.5  W.)                  +  50.258 


Sidereal  time  of  local  0  hrs.  19  57  37.018 

Given  local  sidereal  time  20  11  49.042 


Sidereal  interval  from  noon  0  14  12.024 

Reduction  of  a  sid.  to  a  M.T.  interval  Table  II        —         002.327 


Required  L.  M.  T.  at  instant  of  comparison  0  14  09.697 

Longitude  of  Naval  Academy  West  +    5  05  56.5 


The  G.  M.  T.  at  instant  of  comparison  5  20  06.197 

M.  T.  chronometer  reading  at  comparison  5  24  16 


Error  of  M.  T.  chronometer  on  G.  M.  T.,  fast  4  09.803 


CONVERSION  OF  TIME  393 

194.  Relation  between  apparent  time  and  sidereal  time. — 
From  Art.  173  it  is  seen  that  local  sidereal  time  is  equal  to  the 
true  sun's  right  ascension  plus  the  local  apparent  time,  so  that, 
having  a  given  local  apparent  time,  to  find  the  local  sidereal 
time: 

(1)  Find  the  Greenwich  apparent  time  and  date. 

(2)  Take  out  from  the  Nautical  Almanac  left-hand  column 
of  page  I  (of  the  proper  month),  the  apparent  right  ascen- 
sion and  correct  it  for  the  hours  and  decimals  of  an  hour  of 
G.  A.  T.,  using  the  tabulated  hourly  difference. 

(3)  To  the  local  astronomical  apparent  time  add  the  above 
corrected  apparent  right  ascension.     The  sum,  if  less  than  24 
hours,  will  be  the  local  sidereal  time  (L.  S.T.)  •  if  the  sum  is 
greater  than  24  hours,  reject  24  hours,  and  the  remainder  will 
be  the  L.  S.  T. 

Or  the  following  method  may  be  pursued : 

(1)  Find  the  G.  A.  T.  and  date. 

(2)  For  this  G.  A.  T.  take  out  the  equation  of  time  from 
page  I,  N.  A.     Apply  the  equation  of  time  with  its  proper 
sign  to  the  local  apparent  time  (L.  A.  T.),  obtaining  the  cor- 
responding local  mean  time  (L.  M.  T.)  which  can  be  converted 
into  L.  S.  T.  as  before  explained. 

Ex.  126.— On  January  8,  1905,  in  Long.  135°  15'  E.?  the 
local  apparent  time  is  5h  10m  30s  a.  m.  Find  the  local  side- 
real time. 

h       m       s 

Jan.  7,  local  astronomical  apparent  time  17  10  30 

Longitude  from  Greenwich  East  ( — )  9  01 

Greenwich  apparent  time  Jan.  7,  8  09  30 

h       m          » 

R.  A.  App.  O  at  Greenwich  apparent  noon  Jan.  7,  19  11  49.31 

Corr.  for  G.  A.  T.  (108.949  X  8M6)  1  29.34 

Add  local  astronomical  apparent  time  17  10  30 


Required  local  sidereal  time  12  23  48.65 

195.  Relation  of  time,  hour  angles,  right  ascensions,  and 
a  consideration  of  problems  involving  them. 


394  NAUTICAL  ASTKONOMY 

A  great  many  problems  arise  in  everyday  practical  naviga- 
tion which,  involve  the  consideration  of  hour  angle  and  right 
ascension.  Such  problems  are  readily  solved  if  the  definitions 
of  these  terms  and  of  local  sidereal  time  are  well  understood. 
Some  will  be  illustrated  in  the  following  articles. 

196.  To  find  the  local  mean  time  of  transit  of  a  particular 
heavenly  body  across  the  meridian  of  a  given  place,  the 
longitude  of  the  place,  or  G-.  M.  T.,  being  known. 

(a)  In  case  the  L.  M.  T.  of  transit  of  the  sun  is  desired, 
it  is  only  necessary  to  remember  that  the  instant  of  transit  of 
the  true  sun  is  apparent  noon,  and  at  this  instant  the  equa- 
tion of  time  taken  from  page  I  of  the  Almanac  and  corrected 
for  longitude  (which  is  the  G.  A.  T.  of  the  instant)  is  the 
hour  angle  of  the  mean  sun.  If  the  equation  of  time  is  addi- 
tive to  apparent  time,  the  L.  M.  T.  of  the  sun's  transit  is  the 
equation  of  time,  and  the  local  date  is  the  given  astronomical 
date;  if  the  equation  of  time  is  sub  tractive  from  apparent 
time,  the  L.  M.  T.  of  the  sun's  transit  is  24  hours  —  the  equa- 
tion of  time,  and  the  local  date  is  that  of  the  day  preceding 
the  given  astronomical  day. 

Ex.  127.— January  27,  1905,  in  Long.  52°  30'  W.,  find  the 
local  mean  time  of  upper  transit  of  the  true  sun,  or  of  local 
apparent  noon. 


A  =  G.  A.  T.  of  sun's  transit  =  3h  30™ 
L.  M.  T.  of  local  apparent  noon 
equals  the  equation  of  time, 
or  Jan.  27,  Oh  12^  638.01  (p.  m.) 


Equation  of  Time. 
At  G.  A.  noon  12  51.21 


H.D. 

+  Os.514 
G.A.T.  3h.5 


T ,1-8U    Corr 

Eq.  of  T.         =12  53.01 1 

+  to  Apparent  time. 


Ex.  128.—  April  27,  1905,  in  Long.  52°  30'  W.,  find  the  local 
mean  time  of  the  upper  transit  of  the  true  sun,  or  of  local 
apparent  noon. 


\  =  G.  A.  T.  of  sun's  transit  =  3''  30'" 
L.  M.  T  of  local  apparent  noon 
equals  the  equation  of  time. 

b   m    it 

or  April  27,  (— )    0  02  23.05 

or  April  26,  23  67  36.95 

or  April  27,  11  67  36.95  (a.  m.) 


Equation  of  Time.     I          H.  D. 


At  G.  A.  noon 

Corr. 

Eq.  of  T. 


~~8~l  +08.408 

G.  A.  T.   3h.6 
+  ls.43 


(— )  to  Apparent  time. 


TRANSIT  OF  A  GIVEN  STAR  395 

If  the  G.  M.  T.  of  local  apparent  noon  is  given,  the  equation 
of  time  should  be  taken  from  page  II  of  the  Nautical  Almanac. 

When  any  heavenly  body  is  on  the  upper  branch  of  the 
meridian  of  a  place,  its  right  ascension  is  the  right  ascension 
of  the  meridian,  or  the  local  sidereal  time  (Art.  173),  and  to 
find  the  L.  M.  T.  of  transit  it  is  only  necessary  to  obtain  from 
the  Nautical  Almanac  the  right  ascension  of  the  body  at  that 
instant,  and,  remembering  that  this  is  the  local  sidereal  time, 
reduce  it  to  L.  M.  T.  (Art.  193). 

The  time  of  transit  of  the  sun  across  the  meridian  may  be 
found  in  this  way,  which,  however,  is  a  longer  way  than  the 
method  used  on  page  394. 

(&)  To  find  the  L.  M.  T.  of  transit  of  a  given  star  across 
the  upper  branch  of  a  given  meridian. 

The  American  Ephemeris  and  Nautical  Almanac  contains 
the  apparent  right  ascension  and  declination  of  more  than 
825  of  the  principal  stars  for  the  upper  culmination  at  Wash- 
ington. These  right  ascensions  may  be  taken  as  the  right 
ascensions  for  the  upper  culmination  at  any  other  meridian, 
except  in  the  cases  of  a  few  circumpolar  stars  whose  right 
ascensions  may  be  reduced  by  interpolation  for  differences  of 
longitude,  if  desired. 

In  one  table,  the  mean  places  of  these  stars  are  given  for  the 
beginning  of  the  Besselian  fictitious  year  1912,  that  is,  for  the 
moment  when  the  sun's  mean  longitude  is  280°  (January 
la.006,  1912,  at  Washington).  See  footnote,  page  362. 

In  the  following  table,  the  apparent  places  are  given  for 
every  tenth  upper  transit  at  Washington  for  all  except  the  25 
circumpolar  stars;  for  these  latter  the  apparent  places  are 
given  for  every  upper  transit. 

Knowing  the  name  of  a  star,  its  approximate  right  ascen- 
sion is  found  in  the  former  table,  and  it  may  then  be  con- 
veniently looked  up  in  the  second  table  referred  to  above.  In 
both  tables  the  stars  are  arranged  in  the  order  of  their  right 
ascensions. 


396  NAUTICAL  ASTRONOMY 

Having  the  longitude  of  the  place  and  the  right  ascension 
of  the  star  which  is  the  L.  S.  T.  at  the  instant  of  the  star's 
upper  transit  of  that  meridian,  we  find  the  L.  M.  T.  of  transit 
by  the  method  explained  in  Art.  193. 

Ex.  129.— April  12  (civil  date),  1905,  in  longitude  5h  05m 
56S.5  W.,  find  the  local  mean  time  of  the  upper  transit  of  the 
star  a  Scorpii  (Antares)  whose  E.  A.  is  16h  23m  36S.43. 

An  examination  of  the  approximate  E.  A.  M.  O  and  E.  A. 
of  the  star  shows  the  astronomical  date  to  be  the  llth. 

h     m          a 

April  11,  R.  A.  M.  Q  at  G.  M.  noon  1  16  07.55 

Reduction  for  longitude  West,  Table  III  +  50.258 


Sidereal  time  of  local  0  hrs.  1  16  57.808 

The  given  L.  S.  T.  =  >|<'s  R.  A.  16  23  36.43 


The  sidereal  interval  from  noon  15  06  38.622 

Reduction  of  the  sid.  to  a  M.  T.  interval,  Table  II    (—  )         2  28.531 


The  required  L.  M.  T.  of  -)f' s  transit  April  11,  15  04  10.091 

Or  civil  date  April  12,  (a.  m.)  3  04  10.091 

Ex.  130. — January  5,  1905,  at  the  Naval  Academy,  Annap- 
olis, Md.  (Long.  5h  05m  56S.5  W.),  find  the  75th  meridian 
West  longitude  mean  time  of  the  upper  transit  of  the  star 
a  Canis  Majoris  (Sirius)  across  the  local  meridian. 

h       m          s 

Jan.  5,  R.  A.  M.  Q  at  G.  M.  noon  18  57  38.41 

Reduction  for  longitude  West,  Table  III  +  50.258 


The  sidereal  time  of  local  0  hrs.  18  58  28.668 

The  L.  S.  T.  at  time  of  transit  =  star's  R.  A.  6  40  58.86 


The  sidereal  interval  from  mean  noon  11  42  30.192 

Reduction  of  sid.  to  a  mean  time  interval,  Table  II  (— )       1  55.088 


L.  M.  T.  of  transit  of  Sirius  Jan.  5,  11  40  35.104 

Diff.  of  longitude  of  local  and  75th  meridians  +         5  56.5 


The  75th  meridian  time  of  local  transit  11  46  31.604 


TRANSIT  OF  THE  MOON  397 

(c)  To  find  the  L.  M.  T.  of  transit  of  the  moon  across  the 
upper  branch  of  a  given  meridian. 

In  the  case  of  the  moon  the  L.  M.  T.  of  transit  may  be  found 
from  page  IV  of  the  Nautical  Almanac, where  the  mean  time  of 
the  upper  transit  at  Greenwich  is  given  to  tenths  of  a  minute  ac- 
companied by  a  column  of  differences  for  one  hour  of  longitude. 

These  hourly  differences  express,  for  each  day,  the  mean 
hourly  increase  of  the  moon's  right  ascension.  The  number 
of  minutes  in  this  column  multiplied  by  the  hours  of  longi- 
tude will  give  a  correction,  +  for  West,  ( — )  for  East  longi- 
tude, to  be  applied  to  the  time  of  Greenwich  transit  to  give 
the  L.  M.  T.  of  local  transit. 

In  taking  out  the  time  of  the  meridian  passage  of  the  moon., 
it  must  not  be  forgotten  that  the  result  will  be  astronomical 
time  and  not  civil  time.  When  the  Almanac  time  of  passage, 
after  correction  for  longitude,  gives  a  time  greater  than  12 
hours  of  a  given  astronomical  day,  it  is  plain  that  this  is  not 
the  time  of  passage  on  the  civil  day  of  the  same  date.  Hence, 
if  the  time  of  passage  over  the  meridian*  is  desired  for  a  given 
civil  date,  and  it  is  seen  by  inspection  that  the  tabulated  time 
after  correction  for  the  longitude  will  be  greater  than  12  hours, 
then  it  will  be  necessary  to  take  out  the  Greenwich  time  of 
Greenwich  transit  for  the  day  before  (see  Art.  188). 

Ex.  131. — Find  the  time  of  the  meridian  passage  of  the 
moon  over  the  upper  branch  of  the  meridian  in  longitude 
60°  30'  East  for  January  25  (civil  date),  1905. 

Since  an  inspection  of  page  IV  of  the  Nautical  Almanac 
shows  the  astronomical  time  of  transit  will  be  greater  than 
12  hours,  the  meridian  transit  of  the  preceding  astronomical 
date,  January  24,  must  be  used  in  order  to  make  the  civil  date 
January  25.  „  m  8 

Appro*.  M.  T.    of  transit  of    moon  over  j         ^ 

the  mer.  of  Greenwich  Jan.  24.  ) 

Corr.  or  retardation  for  Long.  East  8  57  A  —  4h.03 

L.  M.  T.  of  local  transit  Jan.  24  15  26  03       Corr.        -8™. 95 

Or  civil  date  Jan.  25,  (a.  m.)  3  26  03 


398  NAUTICAL  ASTRONOMY 

The  above  method  is  sufficiently  accurate  for  all  purposes  of 
everyday  practical  navigation,  but  if  a  more  accurate  time  of 
transit  of  the  moon  is  desired,  find  the  above  L.  M.  T.  of  local 
transit,  apply  the  longitude  to  obtain  the  G.  M.  T.  of  local 
transit,  and  for  this  G.  M.  T.  take  out  the  moon's  right  ascen- 
sion which  is  given  in  the  Nautical  Almanac,  pp.  V-XII,  for 
each  hour  of  G.  M.  T.  with  corresponding  minute  differences. 

The  moon  being  on  the  upper  branch  of  the  meridian,  its 
right  ascension  is  the  L.  S.  T.,  which  can  be  reduced  to 
L.  M.  T.  (Art.  193). 

In  the  solution  on  page  399  will  be  shown  the  method  of 
re-correcting  the  L.  M.  T.  of  transit  of  the  moon  for  the  differ- 
ence between  itself  and  the  approximate  L.  M.  T.  as  found 
above. 

( d)  To  find  the  L.  M.  T.  of  transit  of  a  planet  across  the 
upper  branch  of  a  given  meridian. 

In  the  case  of  a  planet,  the  Nautical  Almanac  gives  the 
G.  M.  T.  of  the  transit  over  the  Greenwich  meridian  for  the 
nearest  tenth  of  a  minute  for  each  day  of  the  year.  The  dif- 
ference of  the  times  for  two  successive  days  will  give  the  daily 
retardation  or  acceleration.  This  divided  by  24  and  the 
result  multiplied  by  the  number  of  hours  of  longitude,  +  for 
West  longitude,  ( — )  for  East  longitude,  will  give  the  retarda- 
tion or  acceleration  to  be  applied  to  the  Greenwich  time  of 
Greenwich  transit  to  give  the  L.  M.  T.  of  local  transit. 

As  in  the  case  of  the  moon,  if  the  sum  of  the  approximate 
time  of  transit  of  the  Greenwich  meridian  and  the  retardation 
(or  acceleration)  is  less  than  12  hours,  the  time  of  the  transit 
of  the  planet  should  be  taken  out  of  the  Nautical  Almanac  for 
the  given  civil  date;  if  that  sum  is  greater  than  12  hours,  the 
time  of  transit  must  be  taken  for  the  day  before  the  given 
civil  day. 


TIME  OF  MOON'S  TRANSIT 


399 


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400  NAUTICAL  ASTRONOMY 

If  a  more  accurate  time  of  transit  of  a  planet  is  desired, 
proceed  as  explained  for  the  case  of  the  moon. 

The  following  example  illustrates  the  finding  of  the  approxi- 
mate as  well  as  of  the  more  exact  time  of  transit. 

Ex.  133. — Find  the  local  mean  time  of  the  meridian  pas- 
sage of  planet  Mars  over  the  upper  branch  of  the  meridian  in 
longitude  52°  30'  W.,  for  April  5,  1905,  civil  date. 

An  inspection  of  page  735,  extracts  from  N.  A.,  1905,  shows 
that  the  astronomical  time  of  transit  is  >  12  hours ;  therefore, 
the  astronomical  date  corresponding  to  the  time  of  transit 
April  5,  civil  date,  is  April  4. 

Acceleration.         B.  A.  M.  Q 
h  m  h  m    s 

Approx.  M.  T.  of  Gr.  Tr.  April  4       14  41.3    For  24h,  4m.l!    At  G.  M.  N.  0  48  31.69 
Acceleration  for  A  =  3^  .5  W         -       0.6    For  IN      0.17  Corr.  G.  M.  T.  2  69.174 


Approx.  L.  M.  T.  of  local  transit     14  40.7    For  A.  W,  0.695  0  61  30.864 

Longitude  West  +  3  30 

G.  M.  T.  of  local  transit )   =  18  10.7 

April  4  (approx.) 

h  m     s  s 

April  4,  R.  A.  of  Mars  at  G.  M.  noon  15  33  18.18       April  4,  H.  D.  —  0.259 

Correction  for  G.  M.  T.  (3*  diff.)  —  5.60       April  5,  H.  D.  -  0.389 

R.  A.  of  Mars  on  meridian  =  L.  S.  T.  15  33  13.58       Change  in  34*>     0.130 

R.  A.  M.  0  corrected  for  G.  M.  T.  0  61  30.864  "        "    l^    0.0054 

L.  M.  T.  of  transit  of  planet  Mars,  April  4,    14  40  41.716  ••        "9.1h     0.049 

or  civil  date  April  5,  (a.  m.)  3  40  41.716     April  4,  H.  D.  —  0.259 

Mean  H.  D.     —  0.308 
G.  M.  T.  18'>  .18 


Correction     —  58.6 

197.  To  find  the  time  of  transit  of  the  moon,  a  planet,  or 
of  a  given  star  across  the  lower  branch  of  a  given  meridian. 

To  find  the  time  of  a  body's  lower  culmination,  the  L.  S.  T. 
is  taken  as  12  hours  plus  the  right  ascension,  or,  what  amounts 
to  the  same  thing,  12  hours  may  be  added  to  the  longitude 
of  the  place.  The  latter  method  is  preferable  when  finding 
the  approximate  times  in  case  of  the  moon  and  planets. 


W.  T.  OF  SUN'S  TRANSIT  401 

Ex.  134. — Find  the  L.  M.  T.  of  the  lower  culmination  of 
the  star  a  Argus  (Canopus)  in  longitude  60°  East  on  April  4, 
a.  m.,  1905. 

In  this  case  (12  hours  +  *'s  K.  A.)  =  18h  21m  508.46 
=  L.  S.  T.  at  the  instant  of  lower  culmination. 

h     m          a 

April  3,  R.  A.  M.  Q  at  G.  M.  noon  0  44  35.13 

Reduction  for  A  60°  E,  Table  III  —  39.426 


The  sidereal  time  of  local  0  hrs.  0  43  55.704 

The  L.  S.  T.  of  lower  culmination  18  21  50.46 


The  sidereal  interval  from  mean  noon  17  37  54.756 

Reduction  of  sidereal  to  a  M.  T.  interval,  Table  II      —        2  53.313 


The  L.  M.  T.  of  lower  culmination  April  3,  17  35  01.443 

Or  civil  time  April  4,  (a.  m.)  5  35  01.443 

198.  To  find  the  watch  time  of  transit  of  a  given  heavenly 
body  across  the  upper  branch  of  a  given  meridian. 

The  simplest  and  most  practical  way  of  observing  the  me- 
ridian altitude  of  a  heavenly  body  is  to  calculate  beforehand 
its  watch  time  of  transit,  and  then  to  observe  the  altitude 
when  the  watch  indicates  that  time. 

(a)  Watch  time  of  sun's  transit. — In  the  case  of  the  sun, 
the  a.  m.  longitude  brought  up  to  noon  by  means  of  the  run 
in  longitude  from  the  time  of  a.  m.  sight  to  noon,  expressed  in 
time,  is  the  G.  A.  T.  of  noon  of  the  given  astronomical  date, 
if  in  West  longitude ;  or,  if  in  East  longitude,  it  is  a  negative, 
or  ( — ),  G.  A.  T.  of  the  given  astronomical  date. 

For  this  G.  A.  T.  take  out  the  equation  of  time,  and  find 
the  G.  M.  T.  of  noon ;  apply  the  chronometer  correction  with 
the  sign  of  application  reversed,  and  get  the  C.  T.  of  noon 
from  which,  by  subtracting  the  C — W,  find  the  watch  time 
of  local  apparent  noon.  Every  navigator  should  do  this  be- 
fore going  on  deck  to  observe  his  meridian  altitude.  Another 


402  NAUTICAL  ASTRONOMY 

way  of  arriving  at  the  same  result  is  to  obtain  from  his  fore- 
noon sight  the  watch  error  on  L.  A.  T.,  and  apply  to  this 
error,  the  difference  in  longitude  for  the  run  from  sight  to 
noon. 

Ex.  135.— April  4,  1905,  in  Long.  85°  30'  W.,  given  the 
C— W  =  5h  52m  05%  chronometer  fast  on  G.  M.  T.  5m  03S.38, 
find  the  W.  T.  of  local  apparent  noon. 

Long.  =  G.  A.  T.  of  local     )  J  ™  "                      Equation  of  time 

apparent  noon  April  4      f  At  G.  A.  H.  j ..,  1Q  g2    n  D  _m 

Equation  of  time  +     3  06.62         April  4     j 

G.  M.  T.  of  local  apparent  noon  5  45  06.62      Corr.               -    4.20    G.  A.  T.  5.7 

Chronometer  fast  on  G.  M.  T.  +     6  03.38      Eq.  of  T.        3  06.62    Corr.  —4.20 

C.  T.  of  local  apparent  noon  5  60  10          +  to  App.  T. 

C— W  6  63  05 

W.  T.  of  local  apparent  noon  11  58  05 

Ex.  136.— January  20,  1905,  in  Long.  132°  15'  E.,  if  the 
C — W  is  3h  17m  30s,  and  the  chronometer  is  slow  on  G.  M.  T. 
6m  19s. 2 9,  what  is  the  watch  time  of  local  apparent  noon  ? 

h  m     •  Equation  of  time 

Long.  =  G.  A.  T.  of  noon  Jan.  20,(-)  8  49  00       At  G  A  N 


or  G.  A.  T.  is  Jan.  19,  15  11  00  T        'n     >  11  05.79  H.D.  +  .739 

Equation  of  time  +       10  69.29       ** 

G.  M.  T.  of  apparent  noon  15  21  69.29  C<>rrectlon  6.5  G.  A.T.-8.8 


Chronometer  slow  (— )  6  19.29 

C.  T.  of  local  apparent  noon  8  15  40 

C-W  8  17  30 


Eq.  of  T.         10  59.29  Corr.  -  6.5 
+  to  App.  T. 


W.  T.  of  apparent  noon  11  58  10 


(&)  Watch  time  of  a  star's  transit. — In  the  case  of  stars, 
the  right  ascension  at  the  instant  of  upper  transit  is  the 
L.  S.  T.  Knowing  the  longitude,  find  the  corresponding 
G.  M.  T.  of  local  transit ;  apply  the  chronometer  correction 
and  C — W  as  in  Exs.  135  and  136  and  get  the  watch  time  of 
transit. 

Remember  that  at  the  instant  of  lower  transit  the  L.  S.  T. 
equals  the  right  ascension  plus  12  hours. 

Ex.  137.— January  10,  1905,  in  longitude  5h  32m  15s  West, 
find  the  watch  time  of  upper  transit  of  the  star  a  Aurigse 


W.  T.  OF  TRANSIT  OF  MOON  OR  PLANET  403 

(Capella)  if  the  C— W  is  5h  35m  10s  and  the  chronometer 
slow  on  G-.  M.  T.  2m  04S.018.  The  star's  E.  A.  =  5h  09m 
41S.68  =  L.  S.  T.  at  transit. 

Jan.  10,  R.  A.  M.  Q  at  G.  M.  noon  19  17  21.19 

Reduction  for  Long.   (5h  32m  15s  W),  Table  III         +         0  54.58 


The  sidereal  time  of  local  0  hrs.  19  18  15.77 

The  given  L.  S.  T.  =  star's  R.  A.  5  09  41.68 


The  sidereal  interval  9  51  25.91 

Reduction  Table  II  —         1  36.892 


L.  M.  T.  of  star's  local  transit  9  49  49.018 

Longitude  West  +    5  32  15 


G.  M.  T.  of  local  transit  15  22  04.018 

Chronometer  slow  on  G.  M.  T.  (— )        2  04.018 


C.  T.  of  star's  local  transit  3  20  00 

C— W  5  35  10 


W.  T.  of  transit  of  star  Capella  9  44  50 

(c)  Watch  time  of  the  transit  of  the  moon  or  a  planet. — 

For  the  moon  or  planets,  find  from  the  Nautical  Almanac 
the  G.  M.  T.  of  local  transit  to  the  nearest  tenth  of  a  minute 
(Art.  188  and  Art.  189),  apply  the  chronometer  correction 
and  C — W  as  above  and  find  the  W.  T.  of  transit. 

199.  To  find  the  hour  angle  of  any  heavenly  body  at  a 
given  time  and  place. — 

(a)  In  the  case  of  the  sun,  the  hour  angle  reckoned  posi- 
tively from  the  upper  meridian  towards  the  West  is  the 
L.  A.  T.  If  the  sun  is  East  of  the  meridian,  the  hour  angle 
is  negative  and  is  equal  to  24  hours — the  apparent  time. 

Having  then  a  given  mean  time  or  sidereal  time,  the  longi- 
tude or  G.  M.  T.  being  known,  the  L.  A.  T.  may  be  found 
by  Art.  190,  Art.  193,  or  Art.  194. 


404 


NAUTICAL  ASTRONOMY 


Ex.   138.— April    10,   a.   m.,    1905,   Long.    129°    30'   45" 
E.,  L.  M.  T.  10h  25m  19s,  find  the  true  sun's  hour  angle. 


Local  ast.  mean  time  April  9, 

Longitude  East 

G.  M.  T.  April  9, 

or  April  10,  (— 10»».  21) 


L.  M.  T.  (astronomical)  April  9, 
Equation  of  time 
L.  A.  T.  =  sun's  H.  A.  April  9. 
or  April  10, 


Equation  of  time 
At  G.M.N.  I     m   s 

April  10     f     1  28.25          — Os.685 
Corr.  +      C.99         —10^.21 

Eq.  of  T.  1  35.24          +  6<>.99 

(-)  to  M.  T. 

h  m  8 

22  25  19 
—    1  35.24 
+  23  23  43.76 
-  1  36  16.24 


Ex.  139.— April  6,  1905,  a.  m.,  Long.  162°  49'  15"  W., 
L.  S.  T.  =  18h  42m  10s,  find  the  H.  A.s  of  mean  and  true 
suns. 


Reduction  for  longitude 
Sidereal  time  of  local  Oh 
The  given  L.  S.  T. 
The  sidereal  interval 
Reduction  to  a  M.  T.  interval 
L.  M.  T.  April  5.     I 
(H.  A.  Mean  Sun)  f 
Longitude  W. 
G.  M.  T.  April  6, 


h    m       a 
)on  0  62  28.24 
+        1  46.99 

Equation  of  time 
(—  )  to  M.  T. 

in        B 

0  54  15.23 
18  42  10 

2  35.86                 H.  D.      —  08.721 
-    3.32                 G.  M.  T.+    4h.6 

17  47  54.77 
-    2  54.951 

2  32.54                 Corr.       -   3s.32 

17  44  59.819 
+  10  51  17 

L.  M.  T.  April  5,            17  44  59.819 
Eq.  of  time                     -    2  32.54 

4  36  16.819 

Vprn5=H'A'°f    174327.279 
or  April  6,                 (-)  6  17  32.721 

Ex.  140.— January  3,  1905,  a.  m.,  in  Long.  150Q  09'  54"  W., 
the  W.  T.  of  obs.  of  the  sun  was  8h  04m  35s,  C— W  10h  07m 
15s,  chronometer  fast  on  G-.  M.  T.  7m  lls.5.  Find  the  true 
sun's  H.  A. 


h    m     e 

W  =  8  04  35 

C-W        10  07  16 
C.  6  11  60 

C.  C.         (— )  7  11.5 
G.M.T.  ' 
Jan 


604  38.5 


Equation  of  time  H.  D. 

At  G.  M.  noon    4  28.38  -f  ls.161 

Corr.  G.  M.  T.  +      7.06  G.  T.     6h.Q8 

Eq.  of  T.  4  35.44  Corr.+  7«.06 

(— )  to  M.  T. 


h  m      s 

G.  M.  T.  6  04  38.5 
Long.  W  10  00  39.6 
L.  M.  T.  20~03~58T<T 
Eq.  of  T.  —435.44 
L.  A.  T.  =  T 

H.A.Q+  19  5923.46 
orII.A.©=-40036.&t 


FINDING  HOUR  ANGLES  405 

(6)  In  the  case  of  the  moon,  a  planet,  or  a  fixed  star,  it 
is  only  necessary  to  find  from  the  given  time,  knowing  the 
longitude  or  Greenwich  mean  time,  the  local  sidereal  time. 
Subtracting  from  this  L.  S.  T.  the  right  ascension  of  the  moon, 
planet,  or  fixed  star,  the  difference,  if  plus,  will  be  the  hour 
angle,  West  of  the  meridian,  of  the  moon,  planet,  or  fixed  star ; 
if  the  difference  is  minus,  it  will  be  the  hour  angle,  East  of 
the  meridian. 

If  the  body  is  the  moon  or  a  planet,  its  right  ascension  is 
corrected  for  the  Greenwich  mean  time  of  the  instant,  but  in 
the  case  of  a  fixed  star,  the  right  ascension,  as  found  in  the 
American  Ephemeris  and  Nautical  Almanac,  is  corrected  for 
the  Washington  mean  time. 

If  the  given  time  is  local  mean  time,  the  right  ascension 
of  the  mean  sun  for  the  Greenwich  instant  must  be  added  to 
it  to  give  the  L.  S.  T.  If  the  given  time  is  local  apparent 
time,  then  the  right  ascension  of  the  true  sun  for  the  Green- 
wich instant  must  be  added  to  the  apparent  time  to  give  the 
L.  S.  T. 

Ex.  IJfl. — About  5  a.  m.  April  22,  1905,  took  an  obser- 
vation of  a  star  a  Aquilse  (Altair),  W.  T.  obs.  4h  55m  20s, 
C— W  lh  12m  00s,  chronometer  fast  on  G.  M.  T.  57m  078.61, 
Long,  by  D.  E.  3°  10'  West.  Find  the  star's  hour  angle. 

h    m      8  B.A.  M.  0  h    m       s 

W  4  55  20  h   m     B         L.  ast.  M.  T.         16  57  33.39 

C— W  1  1200  At  G.  M.  N.     1  55  33.08      Corr.  R.  A.  M.  0    1  68  22.317 

C.  6  07  20  Corr.  G.  M.  T.      2  49.237    L.  S.  T.  18  55  54.707 

C.  C.  —      67  07.61  R.  A.  M.  0      I  68  22.317    Altair's  R.  A.      19  46  09.54 

G  .M.T.  Apr.  21, 17  10  12.39  Altair's  H.  A.(-)  0  60  14.  33 

Long.  W          —      12  40 
L.  M.T.  Apr.  21, 16  57  32.39 


406 


NAUTICAL  ASTRONOMY 


Ex.  U2.— April  10,  1905,  in  longitude  45°  15'  W.,  what 
were  the  hour  angles  of  the  moon  and  the  planet  Mars  at  8h 
06m  14a  p.  m.,  L.  M.  T. 


h   m     B 

L.  ast.  M.  T.  April  10,  8  06  14 
Long,  west             +  3  01  00 

R.  A.  and  H.  A.  of  the  Moon. 

h    m      s 

D's  R.  A.  at  11  h               5  47  50.94  M.  D. 
Corr.  for  7m  .24                          16.04 
D'sR.  A.                           54806.98 
L.  S.  T.                               9  20  14.611 

3 

+  2.2153 

7m.24 
+  168.04 

G.  M.  T.  April  10,     11  07  14 
or  llh.12 
h    m  s 

L.  ast.  M.  T.  Apr.  10,  8  06  14 
R.A.M.QatG.M.N.    11211 
Bed.  for  G.  M.  T.              1  49.611 

D'SH.  A.                         +33207.631 
or  West  3  32  07.631 

R.  A.  and  H.  A.  of  Mars. 

Local  sidereal  time    9  20  14.611 

h    m        s 

R.  A.  of  Mars  April  10,  15  30  44.17   H.  D 
Corr.  for  G.  M.  T.               —11.65 

8 

—  1.048 
11M2 
lla.65 

R.  A.  of  Mars                 15  30  32.52 
L.  S.  T.                               9  20  14.611 

H.  A.  of  Mars              —  6  10  17.909 
or  East    6  10  17.909 

Ex.  143.— On  April  5,  1905,  in  longitude  34°  52'  30"  W. 
the  H.  A.  of  the  true  sun  is  +  3h  10m  30s,  find  the  H.  A.  of 
the  vernal  equinox  and  stars  Sirius  (a  Canis  Majoris)  and 
Achernar  (a  Eridani). 


Local  astronomical  apparent  time  April  5, 
Longitude  from  Greenwich  West 

Greenwich  apparent  time  April  5, 

R.  A.  App.  Q  at  G.  A.  N.  April  5  (p.  I) 
Correction  for  G.  A.  T.  (9M25  X  5h.5) 
Local  astronomical  apparent  time 

L.  S.  T.  equals  the  H.  A.  of  the  vernal  equinox 


L.  S.  T. 
R.  A. 


Sirius 


4  06  42.13 
6  40  57.81 


H.A.  >|<  Sirius     (—  )  2  34  15.68 


L.  S.  T. 

R.  A.  >|<  Achernar 

H.  A.  >|<  Achernar 


3  10  30 

f  2  19  30 

5  30  00 

h     m          8 

0  55  21.94 
0  50.19 

3  10  30 

4  06  42.13 

4  06  42.13 

1  34  07.75 

+T32  34.38 


Attention  is  called  to  the  fact  that  for  the  G.  A.  T.  the 
equation  of  time  might  have  been  taken  from  page  I  and  ap- 
plied to  the  G.  A.  T.  to  obtain  G.  M.  T.  and  then  the  L.  S.  T. 
found  from  the  G.  M.  T.  as  in  previous  examples. 


H.  A.  OF  POLARIS 


407 


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408  NAUTICAL  ASTRONOMY 

200.  To  find  the  local  time  when  the  hour  angle  of  a  par- 
ticular heavenly  body  and  the  Greenwich  time  are  known, 
or  when  the  hour  angle  of  a  fixed  star  and  the  longitude  are 
known. 

In  the  case  of  the  sun,  its  hour  angle  reckoned  westward 
is  the  L.  A.  T.  of  the  given  astronomical  day ;  if  the  sun  is 
East  of  the  meridian,  the  L.  A.  T.  is  24  hours  —  the  hour 
angle,  and  the  date  is  that  of  the  preceding  astronomical 
day.  This  L.  A.  T.  may  be  reduced  to  mean  or  sidereal  time 
(Art.  190  or  Art.  194),  as  required,  the  Greenwich  time 
or  longitude  being  known. 

In  the  case  of  any  other  heavenly  body,  find  its  right 
ascension  for  the  Greenwich  instant.  This,  added  algebraic- 
ally to  the  hour  angle,  will  give  the  L.  S.  T.  Subtracting 
from  this  the  right  ascension  of  the  sun  (true  or  mean), 
taken  from  the  Nautical  Almanac  for  the  Greenwich  instant, 
the  remainder  will  be  the  hour  angle  of  the  sun  (true  or 
mean),  and  the  hour  angle  will  be  the  local  time  (apparent  or 
mean),  or  24  hours  minus  the  local  time  (apparent  or  mean), 
according  as  the  hour  angle  is  -j-  or  ( — ) . 

If,  in  finding  the  hour  angle  by  subtracting  the  right  ascen- 
sion from  the  L.  S.  T.,  it  is  seen  that  the  L.  S.  T.  is  less  than 
the  right  ascension,  and  it  is  desired  to  express  the  hour  angle 
positively,  add  24  hours  to  the  L.  S.  T.  before  performing 
the  subtraction. 

Ex.  145.— April  16,  1905,  G.  M.  T.  10h  18m,  the  moon's 
hour  angle  is  2h  30m  East  of  the  meridian  of  a  certain  place. 
Find  the  L.  M.  T. 


Right  Ascension  of  the  Moon,  L.  S.T.  and  L.  M.T. 


R.  A.  M.  0 


April  16  at  IQh  H  20  03.43          M.D.  +  2«.3708 


April  16 
at  G.  M.  noon  f   1  35  50.32 


Corr.  for  18  minutes       +42.674  18«n 

Corr.  for  G.M.  T.      141. 


Corrected  R.  A.  C      11  20  46.104        Corr.  +  428.674 
Moon '8  H.  A.  East  —  2  30  00 
L.  S.  T.  8  50  46.104 

Corrected  R.A.M.Q    1  37  31.842 
L.  M.  T.  April  16,        7  13  14.263 


R.  A.  M.  0  1  37  3U 


STARS  TO  CROSS  MERIDIAN  409 

Ex.  11+6. — April  9,  astronomical  time,  1905,  at  a  given  in- 
stant the  hour  angle  of  the  star  a  Canis  Minoris  (Procyon) 
at  Annapolis,  Md.,  was  3  hours  West  of  the  meridian.  At  the 
same  instant  the  hour  angle  of  the  star  a  Leonis  (Regulus) 
was  1  hour  East  of  a  second  meridian.  Find  the  L.  M.  T. 
at  each  meridian. 

h    m        s  h    m        3 

R.  A.  of  star  Procyon  7  34  20.18  R.  A.  of  star  Regulus  10  03  19.95 

H.  A.       do.        do.                       +  3  00  00  H.  A.    do.         do.   —    1  00  00 

L.  S.  T.  (at  Annapolis)  10  34  20.18  L.  S.  T.  (2d  meridian)   9  03  19.95 

April  9,  R.A.M.  0  at  GL  M.  noon  1  08  14.45  L.  S.  T.  (Annapolis)     10  34  20.18 

Reduction  for  Long.  W,  Tab.  Ill  4         60.258  Diff .  longitude    —        1  31  00.23 

Sidereal  time  local  Qh  1  09  04.708  L.  M.  T.  at  Annapolis    9  23  42.868 

The  given  L.  S.  T.  10  34  20.18  L.  M.  T.  (2d  meridian)  7  52  42.638 

Sidereal  interval  9  25  15.472  or  April  9,  (p.  m.)        7  52  42.638 

Reduction  Tab.  II  -    1  32.604 

L.  M.  T.  at  Annapolis  9  23  42.868 

or  civil  time  April  9,  (p.  m.)  9  23  42.868 

201.  Given  two  mean  times  or  two  apparent  times  at  a 
given  place,  to  find  what  bright  stars  will  cross  the  upper 
branch  of  the  meridian  between  those  two  times. 

Since  the  right  ascension  of  a  body  on  the  meridian  is  the 
right  ascension. of  the  meridian,  or,  in  other  words,  the  L.S.T., 
it  is  only  necessary  to  find  the  local  sidereal  times  correspond- 
ing to  the  two  given  times.  Any  star  whose  right  ascension 
lies  between  the  two  L.  S.  Ts.  thus  determined  will  cross  the 
upper  branch  of  the  meridian  between  the  two  given  times,, 
and  any  star  whose  right  ascension  lies  between  the  two  local 
sidereal  times  increased  by  12  hours  of  sidereal  time  will 
pass  the  lower  branch  of  the  meridian  between  the  two  given 
times. 

The  "  mean  place  catalogue  "  of  stars  in  the  Nautical  Al- 
manac is  the  more  convenient  one  to  use  for  this  purpose. 

The  visibility  of  a  star  at  the  time  of  its  transit  over  any 
meridian  will  depend  on  the  latitude  of  the  place  and  the  de- 
clination of  the  star,  which  determine  whether  the  star  is 
above  or  below  the  horizon. 


410  NAUTICAL  ASTRONOMY 

At  sea,  the  mean  or  apparent  time  of  transit  of  a  heavenly 
body  for  a  certain  meridian  is  obtained  with  the  idea  perhaps 
of  observing  the  body's  altitude  when  on  the  meridian.  It 
must  not  be  forgotten  that  the  ship's  clock  was  regulated  to 
apparent  time  at  noon,  and  that  the  navigator  must  learn 
his  watch  error  on  the  local  time  of  the  meridian  over  which 
he  is  to  observe  a  transit.  If  his  watch  was  correct  at  noon, 
it  will  be  too  fast  on  the  local  time  of  a  meridian  to  the 
westward,  too  slow  on  the  local  time  of  a  meridian  to  the 
eastward  of  the  noon  meridian  by  four  minutes  of  time  for 
each  degree  difference  of  longitude.  It  would  be  well,  how- 
ever, for  the  navigator  to  carry  a  watch  regulated  to  G.  M.  T., 
and,  having  found  the  Greenwich  mean  time  corresponding  to 
the  required  transit,  to  observe  by  that  watch. 

Ex.  147. — What  stars  of  a  magnitude  greater  than  the 
second  magnitude  crossed  the  upper  branch  of  the  meridian  of 
Annapolis,  Md.,  above  the  visible  horizon,  between  the  hours 
of  8  p.  m.  and  12  midnight  t)f  75th  meridian  West  longitude, 
mean  time,  January  18,  1905. 

h   m     s  h   m      s 

75th  meridian  mean  time  8  00  00  12  00  00 

Long,  of  75th  meridian  W  5  00  00  5  00  00 


G.  M.  T  Jan.  18,  13  00  00  17  00  00 

Longitude  of  Annapolis  West  —  5  05  56.5  —  5  05  56.5 


Local  astronomical  mean  times  7  54  03.5  11  54  03.5 

R.  A.  M.  0  Jan.  18  at  G.  M.  noon  19  48  53.64  19  48  53.64 

Correction  for  G.  M.  T.  +          208.134  +      247.56 


L.  S.  Ts.  =  limits  of  R.  As.  3  45  05.274  7  45  44.70 

All  stars  of  a  greater  magnitude  than  the  second  whose 
right  ascensions  fall  between  the  above  limits  and  whose 
South  declination  is  <51°  01'  07"  S. 

a  Tauri.  e  Orionis.  8  Canis  Majoris. 

a  Aurigae.         a  Orionis.  a2  Geminorum. 

(3  Orionis.        a  Canis  Majoris.        a  Canis  Minoris. 

13  Tauri.          €  Canis  Majoris.         ft  Geminorum. 


CHAPTER  XV. 
CORRECTIONS  TO  AN  OBSERVED  ALTITUDE. 

202.  The  observed  altitude  of  a  heavenly  body  above  the 
sea  horizon,  at  a  given  place,  is  the  altitude  of  the  body  as 
indicated  by  the  reading  of  the  sextant  with  which  the  obser- 
vation was  made,  after  correction  for  the  index  error  (I.  C.) 
previously  explained. 

The  true  altitude  of  the  body,  at  the  given  place,  is  the 
altitude  of  its  center  observed  above  the  celestial  horizon/  the 
eye  of  the  observer  supposed  to  be  at  the  center  of  the  earth. 
This  point  is  selected  as  the  common  point  to  which  to  refer 
observations  made  at  the  surface,  when  combining  them  with 
the  tabulated  elements  from  the  Nautical  Almanac,  in  the 
solution  of  the  astronomical  triangle. 

To  reduce  an  observed  altitude  of  a  heavenly  body  to  a 
true  altitude  it  is  necessary  to  apply  the  following  correc- 
tions: Dip,  refraction,  parallax,  semi-diameter.  Theoretic- 
ally they  should  be  applied  in  the  above  order;  following  that 
order  would  give: 

after  applying  dip, 

(1)  the  apparent  altitude  of  the  limb; 

after  applying  refraction  and  parallax, 

(2)  the  true  altitude  of  the  limb; 

after  applying  semi-diameter, 

(3)  the  true  altitude  of  the  center. 

When  an  artificial  horizon  is  used,  the  observed  and  ap- 
parent altitudes  are  the  same ;  in  other  words,  there  is  no  cor- 
rection for  the  dip.  As  already  explained  under  the  head 


412  NAUTICAL  ASTRONOMY  ^ 

of  artificial  horizon,  when  a  body  is  observed,  the  artificial 
horizon  being  used,  the  reading  of  the  sextant  is  first  cor- 
rected for  I.  C.,  and  the  corrected  reading  divided  by  2  to 
get  what  is  known  as  the  observed  altitude. 

In  case  of  fixed  stars,  owing  to  their  great  distances,  the 
semi-diameter  and  parallax  are  inappreciable,  so  that  the  only 
corrections  to  be  applied  are  I.  C.,  dip,  and  refraction. 

In  case  of  planets,  for  sea  observations,  parallax  and  semi- 
diameter  may  be  disregarded;  however,  if  the  observation  is 
made  with  a  telescope  so  powerful  that  the  limb  can  be  dis- 
tinguished, the  semi-diameter  should  be  applied. 

For  refined  observations  ashore,  both  these  corrections 
should  be  applied. 

For  the  ordinary  sea  observations  of  a  planet,  it  will  be 
sufficient  to  correct  the  altitude  for  I.  C.,  dip,  and  refraction. 

In  ordinary  nautical  practice,  it  is  unnecessary  to  follow 
the  theoretical  order,  except  that  in  the  case  of  the  moon,  it 
is  essential  to  find,  first,  the  apparent  altitude  of  the  moon's 
center,  and  for  this  to  take  out  the  correction  for  parallax  and 
refraction  combined.  (Bowditch,  Table  24.) 

203.  Refraction. — It  is  a  fundamental  law  of  optics  that 
a  ray  of  light,  when  passing  obliquely  from  one  medium  into 
another  of  different  density,  is  bent  towards,  or  from,  a  normal 
to  the  separating  surface  at  the  point  of  entrance,  according 
as  it  passes  from  a  lighter  into  a  denser  medium,  or  the 
reverse. 

The  ray  before  entering  the  second  medium  is  called  the 
incident  ray,  after  entering  it  the  refracted  ray.  The  inci- 
dent ray  makes  with  the  normal  what  is  called  the  angle  of 
incidence,  the  refracted  ray  makes  with  the  normal  the  angle 
of  refraction,  and  the  difference  between  these  two  angles  is 
called  the  refraction. 

Astronomical  refraction.— A  ray  of  light  from  a  heavenly 
body  must  pass  through  the  atmosphere  before  reaching  the 
observer. 


REFRACTION 


413 


The  earth's  atmosphere  may  be  considered  as  formed  of 
concentric  spherical  strata,  that  nearest  the  surface  of  the 
earth  being  of  greatest  density,  and  each  succeeding  stratum 
decreasing  in  density  as  its  distance  from  the  surface  in- 
creases till  the  upper  limit  of  the  atmosphere  is  reached  at 
a  height  of  perhaps  50  miles  from  the  surface. 

If  the  space  between  the  upper  limit  of  the  atmosphere  and 
a  star  be  regarded  as  a  vacuum,  or  filled  with  a  medium  which 
exerts  no  sensible  effect  on  the  direction  of  a  ray  of  light,  its 


FIG.  101. 


path  will  be  a  straight  line  till  it  meets  the  upper  limit  of 
the  atmosphere. 

At  this  upper  limit,  the  effect  of  refraction  is  very  small, 
but,  as  the  ray  continuously  passes  through  the  atmosphere 
whose  density  increases  by  insensible  degrees  from  stratum 
to  stratum  of  infinitely  small  thickness,  its  path  is  a  curve 
concave  to  the  surface  of  the  earth ;  the  plane  of  its  path  being 
in  the  plane  of  the  normals  which  meet  at  the  center  of  the 
earth. 

The  last  direction  of  a  ray,  or  that  at  which  it  enters  the 
eye  of  the  observer,  is  in  a  tangent  to  the  curve  at  this  point 


414  NAUTICAL  ASTRONOMY 

and  indicates  the  direction  in  which  a  body  appears  to  the 
observer;  so  it  is  apparent  that  the  effect  of  the  bending  of 
the  rays  is  to  apparently  increase  the  altitude  of  the  body 
without  altering  its  azimuth.  Astronomical  refraction,  then, 
which  is  the  difference  of  direction  between  the  ray  that 
enters  the  eye  of  an  observer  and  of  the  same  ray  before 
entering  the  atmosphere,  making  as  it  does  an  altitude  appear 
greater  than  it  really  is,  must  be  subtracted  from  an  observed 
altitude  of  a  body. 

The  ray  from  a  star  8f  entering  the  atmosphere  at  B 
(Fig.  101)  is  bent  into  the  curve  BA.  The  observer  at  A 
apparently  sees  8  in  direction  AS'.  The  angle  EBS  is  the 
angle  of  incidence;  ZAS',  the  angle  of  refraction;  and  the 
ratio  of  their  sines  is  a  constant  at  a  given  place  for  a  given  at- 
mospheric condition.  Refraction  equals  EBS  —  EDS',  be- 
cause the  angle  between  AS'  and  B8  is  equal  to  the  difference 
of  the  angles  that  these  lines  make  with  any  straight  line  cut- 
ting both. 

The  refraction  for  a  mean  state  of  the  atmosphere,  that 
of  a  height  of  barometer  of  30  inches  and  temperature  of 
50°  F.,  can  be  found  in  Table  20A,  Bowditch. 

A  rise  of  temperature,  or  a  fall  of  barometer,  indicates  a 
decrease  of  density  of  the  atmosphere  and  hence  a  diminution 
of  refraction.  A  fall  of  temperature,  or  a  rise  of  barometer, 
would  indicate  the  reverse  (see  Tables  21  and  22,  Bowditch). 
In  Table  20B,  Bowditch,  will  be  found,  in  the  case  of  the 
sun  only,  the  value  of  combined  parallax  and  refraction. 

Refraction  is  zero  when  the  body  is  in  the  zenith,  about 
36'  when  it  is  in  the  horizon,  and  for  intermediate  altitudes 
may  be  said  to  vary  as  the  tangent  of  the  zenith  distance  of 
the  body,  provided  the  zenith  distance  does  not  exceed  80°. 

Owing  to  the  irregularity  of  refraction  at  low  altitudes  it 
is  advisable  not  to  observe,  at  sea,  altitudes  of  less  than  10°. 

The  oval  form  of  the  sun  and  moon  after  rising  and  before 


PARALLAX 


415 


setting  is  due  to  the  difference  of  refraction  for  the  altitudes 
of  the  lower  and  upper  limbs. 

Kefraction  affects  the  dip,  decreasing.it  by  about -j^th  of  the 
whole. 

204.  Parallax. — In  general,  parallax  may  be  denned  as  a 
change  in  direction  of  an  object  due  to  a  change  of  the  point 
of  view.  In  astronomical  observations,  the  observer  is  on  the 
surface  of  the  earth,  and  it  is  desired  to  reduce  observations 
to  what  they  would  be  if  the 
observer  were  at  the  center  of 
the  earth.  It  is  by  the  appli- 
cation of  parallax  that  obser- 
vations are  so  reduced. 

Geocentric  parallax  is  the 
angle  at  the  body  subtended  by 
that  radius  of  the  earth  which 
passes  through  the  observer's 
position  at  the  surface.  When 
the  heavenly  body  is  in  the 
horizon  at  H  (Fig.  102),  this 
angle  has  its  greatest  value  and 
CAR  is  a  right  angle.  Let-  FIG.  102. 

ting  this  angle,  called  the  hori- 
zontal parallax,  be  represented  by  P,  the  earth's  equatorial 
radius  by  R,  the  distance  of  the  body  by  d,  we  have 


sin  P  = 


R 

d 


The  value  of  P  from  this  formula  is  given  in  the  Nautical 
Almanac  for  the  sun,  moon,  and  planets. 

Parallax  in  altitude. — When  a  heavenly  body  is  observed 
in  any  position  other  than  in  the  horizon,  the  parallax  to  be 
applied  is  known  as  parallax  in  altitude. 


416 


NAUTICAL  ASTRONOMY 


In  triangle  CAS,  p  is  the  parallax  in  altitude,  ZAS  the 
apparent  zenith  distance  =  z'  =  90°  -  -  h', 

ZC8  the  true  zenith  distance  of  body  =  z, 
d  the  distance  of  the  body,  and  we  have 
sin 
sin  z' 
sin  p  =  sin  P  cos  h'-. 

Since  p  and  P  are  small  angles,  they  are  proportional  to 
their  sines;  therefore, 

p  =  Pcosh'.  (143) 

Parallax  is  additive  to  the  observed  altitude. 

205.  Dip  of  the  horizon. — The  visible  sea  horizon  is  the 
small  circle  where  tangents  from  the  observer's  eye  meet  the 
sea.  The  sensible  and  celestial  horizons  have  already  been 
defined  (Art.  138). 

The  dip  of  the  horizon  is  the  angular  depression  of  the 

visible  below  the  celestial  hori- 
zon, and  is  due  to  the  elevation 
of  the  observer's  eye  above  the 
surface. 

In. Fig.  103,  let  BR  be  a 
portion  of  the  earth's  surface, 
C  the  center,  CO  a  radius  pro- 
longed to  A,  the  eye  of  an  ob- 
server; CB  and  CB'9  radii 
making  angles  of  90°  respec- 
tively with  AB  and  AB',  tan- 
gents to  the  surface.  If  this 
figure  be  revolved  about  AC, 
HH'  will  generate  a  plane  par- 
allel to  the  celestial  horizon  and  either  A B  or  AB'  will  gener- 
ate a  cone  tangent  to  the  earth  at  the  visible  horizon. 

Letting  R  be  the  earth's  radius  in  feet,  h  the  height  of 


FIG.  103. 


DIP  417 

observer's  eye  in  feet, and  D  the  dip, since  D  =  HAB  =  ACS, 
we  have 

R 
cos  D  =  R   I   ^ ;  but  cos  D  "  *  —  2  sin2  -J  D, 


sn        = 


/  7}"\2 
As  Z)  is  small,  sin2  ^  D"  =  (        )  sin2  1",  and  as  h  is  very 

V  *  / 

small  in  comparison  with  R,  justifying  the  assumption  that 
R  -f-  h  is  sensibly  equal  to  R,  we  have 

/)"  -        1         l^h 
-sml"YlT 

The  value  of  the  mean  radius  in  feet  being  20,902,433  feet, 
Z>"  =  63".803  yh,  (144) 

D'  =     1'.063  V^  (145) 

However,  the  value  of  the  dip,  as  found  above,  is  affected 
by  refraction  which  raises  the  visible  horizon,  increases  the 
distance  at  which  an  object  in  the  horizon  can  be  seen,  and 
lessens  the  dip,  so  that  when  the  effect  of  refraction  is  to  be 
considered  a  change  must  be  made  in  the  formula. 

For  a  mean  state  of  the  atmosphere,  barometer  30  inches, 
thermometer  50°  F.,  it  has  been  computed  that  the  value  of 
the  dip,  considering  refraction,  is  given  by  the  formula, 

=  58".801  ^  h  (146) 


sin  J. 
Dr'    =  0'.98  \/  h  (147) 

Application  of  dip.  —  Table  14,  Bowditch,  gives  the  dip  for 
various  heights  of  the  eye,  computed  so  as  to  allow  for  the 
effect  of  the  refraction  of  the  atmosphere  under  normal  con- 
ditions. Dip  is  one  of  the  corrections  to  be  applied  to  an 
observed  altitude  of  a  heavenly  body  to  obtain  the  true  alti- 
tude, and  is  subtractive  to  the  observed  altitude,  as  the  visible 
horizon  is  below  the  celestial  horizon. 


418  NAUTICAL  ASTRONOMY 

Error  of  dip. — The  position  of  the  visible  horizon,  and 
hence  the  amount  of  dip,  depends  on  the  relative  temperature 
of  sea  and  air.  The  horizon  is  depressed  below  its  •  mean 
position,  and  the  dip  is  increased  over  the  tabulated  amount, 
when  the  sea  is  warmer  than  the  air ;  the  reverse  is  true,  when 
the  air  is  warmer  than  the  sea. 

Hence  it  is  easily  understood  that  tabulated  dip  for  given 
conditions  may  be  in  error,  and  that  this  error  will  affect  all 
altitudes  observed  under  those  conditions.  The  error  of  posi- 
tion thus  caused  may  be  considerable,  especially  in  the  Eed 
Sea  and  in  regions  of  the  Gulf  stream.  For  this  reason,  the 
navigator  must  be  cautious  and,  as  experience  shows  that  the 
error  decreases  with  the  height  of  the  observer's  eye,  it  would 
be  well  for  him  to  observe  from  elevated  positions. 

Chauvenet  gives  the  following  formula  from  which  to  find 
a  correction,  always  subtractive  to  Dr" : 

24021"  (*  — *0)     , 
Corr.  =  -     1-Q —  when 

Dr 

t  is  the  temperature  of  air,  and  t0  that  of  water,  using  a 
Fahrenheit  thermometer. 

206.  To  find  the  distance  of  the  visible  horizon  for  a  given 
height  h  of  observer's  eye. — It  has  been  seen  that  refraction 
reduces  the  angle  of  dip  and  increases  the  distance  of  the 
visible  horizon,  so  that  the  distance  of  the  visible  horizon 
from  an  altitude  In,  when  the  dip  is  affected  by  refraction,  may 
be  considered  to  be  the  same  that  it  would  be  from  an  alti- 
tude h  +  x,  provided  there  was  no  effect  of  refraction.  Let- 
ting d  be  the  distance  of  the  visible  horizon  from  the  height 
of  eye  of  h  +  &  feet  and  as  before  R  the  radius  or  the  earth 
in  feet,  refraction  not  being  considered, 

d=  ^(R  +  h  +  x)2  —  R2 


2Rh       2Rx       2hx  — 


RANGE  OF  VISIBILITY  419 

Since  (h  -}-  x)2  is  very  small  in  comparison  with  2R  (h  +  x), 
let  d  =  \/2R  (h  +  x)  y  a;  is  a  side  of  a  triangle  which,  with- 
out appreciable  error,  may  be  considered  as  right  angled,  and 
the  angle  opposite  x  may  also,  without  appreciable  error,  be 
taken  as  D  —  Dr  ; 

therefore,  x  —  d  sin  (D  —  Dr), 


x  =  sin  (D  —  Dr)  V2#  (h  +  x)  ; 
but  D  —  Dr  =  5".002 


hence  x  =  5.002  \A  V^5  (ft  +  x)  sin  1" 
z2  =  50.045  (h2  +  for)  sin2  I" 

sin2  1"  +  (25.02)  *R2h2  sin4  1" 

=  50.045ft2  sin2  1"  +  (25.02)  252ft2  sin4  1" 
a  _25.025ft  sin2  1" 

=  ±  A  V50.045  sin2  V  +  (25.02)  252  sin*  I" 

x  =25.Q2Rh  sin2  1"          __ 

±  h  V50.04B  sin2  l/r  +  (25.02)  2E2  sin4  1" 

Whence,  since  R  =  20902433  feet, 
25.025  sin2  1"  =  .01229. 
50.045  sin2  1"  =  .  02458  1  = 

(25.02)252  sin4  l/r  =  .00015  J 


x  =  .16955&,  and  d  —  y2R(h  +  z)  =  -\/2.3391Rh  in  feet. 
d  (in  nautical  miles)  - 


=  1.15V*- 

c?=  1.15V^>  <^  in  nautical  miles  1  ,    .      „    ,          (148) 
d  =  1.324 V^,  d  in  statute  miles  J  (149) 

207.  Kange  of  visibility  at  sea. — If  an  observer  whose  eye 
is  at  A,  height  h  feet,  sees  in  his  horizon  at  T  (Fig.  103) 
a  light  or  object  of  known  height  h',  then  since 

AB'  =  1.15V* 
B'T  =  1.15  V*', 


420 


NAUTICAL  ASTRONOMY 


the  distance  of  the  light  in  nautical  miles  will  be 


and  in  statute  miles, 

d  =  1.324  (  yh  + 
Table  6,  Bowditch,  gives  the  distance  of  visibility  of  objects  at 
sea  in  both  nautical  and  statute  miles  for  a  given  height  of 
eye.  Entering  this  table  with  heights  of  observer  and  object, 
respectively,  the  sum  of  the  corresponding  distances  will  be 
the  distance  of  the  object  from  the  observer. 

Ex.  148.  —  A  light  121  feet  above  the  level  of  the  sea  is 
just  visible  from  a  bridge  of  a  steamer  49  feet  above  the  water. 
Eequired  the  distance  of  the  light  in  nautical  miles. 
d  =  1.15  (  Vm  +  V49)  =  1.15  (11  +  7) 
=  1.15  X  18  =  20.7  miles. 

208.  To  find  the  dip  or  de- 
pression of  a  point  nearer  than 
the  horizon,  as  of  a  land  hori- 
zon. —  In  Fig.  104,  x  represents 
the  height  of  an  observer,  hav- 
ing in  sight  a  shore  horizon  B, 
which  corresponds  to  the  visible 
sea  horizon  of  a  height  y  on 
the  perpendicular  through  the 
eye  of  the  observer,  and  d 
equals  the  known  distance  of 
B.  But  by  a  previous  article 

d  =  1.15  Vy  (in  nautical  miles),  therefore  V#  —  T^TK>  ^u^ 
the  dip  at  height  y  after  correction  for  refraction  is  Dr" 

=  58".801  yy  =  58"-8Qld  =  51".13d. 
1.15 

The  refracted  rays  Bx  and  By  make  with  each  other  a 
small  angle  which  represents,  without  appreciable  error,  the 


FIG.  104. 


SHORE  HORIZON  421 

difference  of  dip,  or  depression  of  B,  as  seen  from  the  heights 
#  and  y;  letting  </>"  represent  this  angle  in  seconds  of  arc, 
since  Bxy  is  nearly  right  angled  at  y,  we  have 

Tan  <f>"  =.  -  x      y—  and,  as  </>"  is  a  small 


x  —  y 
6080.27  d  tan  1"  =  6080.27d  tan  1" 


x  d 


6080.27^  tan  1"       (1.15)2  6080.27  tan  1" 
|_25".651d. 

Eemembering  what  <j>"  is,  and  knowing  that  the  dip  at 
height  y  equals  Dr"  =  51". 13d,  to  find  the  dip  of  B  at  height 
x,  represented  by  Drx",  we  have  only  to  add  <£"  to  Dr"  at 
height  y. 

Therefore  Drx"  =  25".479d  +  33".924  |. 

For  practical  purposes  it  is  only  necessary  to  use  the  for- 
mula to  the  nearest  tenth, 
or  f  x  in  feet, 


in  nautical  miles, 


n  _  9K//  K/7  .r  QQ//  Q  a;         n  nauca  , 

-  3  '9  31  DrJ'  dip  in  seconds  of  arc, 


(160> 


corrected  for  refraction. 

Shore  horizon.—  When  sailing  near  shore,  or  when  in  a 
harbor  at  anchor,  an  observer  may  be  forced  to  use  an  alti- 
tude from  a  shore  horizon.  The  dip  may  be  calculated  by  the 
above  formula,  or  taken  out  of  Table  15,  Bowditch. 

209.  Apparent  semi-diameter.  —  The  apparent  semi-diameter 
of  a  body  is  the  angle  subtended  by  its  radius  at  the  place  of 
the  observer,  and  for  the  same  body  varies  with  the  distance 
of  that  body  from  the  observer.  The  value  given  in  the 
Nautical  Almanac  is  the  angle  at  the  center  of  the  earth  sub- 
tended by  the  radius  of  the  body. 


422 


NAUTICAL  ASTRONOMY 


At  sea,  in  sextant  observations  of  the  sun  and  moon,  the 
upper  or  lower  limb  is  brought  into  contact  with  the  sea 

horizon;  in  observations  on 
shore,  when  using  an  artifi- 
cial horizon,  the  opposite 
limbs  of  direct  and  reflected 
images  are  made  tangent  to 
each  other. 

Since  the  altitude  of  the 
center  of  the  body  is  re- 
quired, the  angular  semi-di- 
ameter of  the  heavenly  body 
observed  must  be  applied, 
plus  or  minus,  according  as 
the  altitude  observed  was 
that  of  the  lower  or  upper 
limb ;  and  as  the  observation 
of  the  limb  is  reduced,  by 
FlG  105.  the  application  of  parallax, 

to  what  it  would  be  if  taken 

at  the  center  of  the  earth,  it  is  necessary  to  find  the  apparent 
semi-diameter  of  a  heavenly  body  as  it  would  be  seen  at  that 
point. 

In  Fig.  105,  let  M  be  the  body, 
}i   its  apparent  altitude, 

its  true  altitude,  or  90°  —  ZCM. 
its  apparent  zenith  distance, 
its  true  zenith  distance, 
its  distance  from  A, 

its  distance  from  the  center  of  the  earth, 
8  =  MCB,  apparent  S.  D.,  as  viewed  from  C,  the  center 

of  the  earth, 

S'  —  MABf,  apparent  S.  D.,  as  viewed  from  A,  the -ob- 
server's position  on  the  surface, 


li 

zr 
z 

at 

d 


APPARENT  S.  D.  423 

R  =  CA  =  earth's  radius, 

r  =  MB  =  MB'  =  linear  radius  of  body. 

From  the  right-angled  triangle  MOB,  sin  8  =  T-T. 

When  the  body  M  is  in  the  horizon  of  A,  AM  and  CM  are 
sensibly  equal  and,  hence,  the  angle  8  is  called  the  horizontal 
semi-diameter. 

7?  7? 

It  has  been  shown  that  sin  P  =  -=-  or  d  =   .     „. 

d  sin  P 

Therefore,  sin  8  =-5-  sin  P. 

Since  8  and  P  are  small,  they  are  proportional  to  their 
sines,  hence  8  —  ~  P. 

~  is  a  ratio,  constant  for  any  particular  body,  and,  repre- 
senting it  by  C,  we  have 

log  8  =  log  C  +  log  P.  (151) 

For  the  moon,  ^  =  .272,  so  that  having  the  moon's  hori- 
zontal parallax,  its  semi-diameter  may  be  gotten  by  multiply- 
ing it  by  .272 ;  however,  it  is  just  as  easy  to  take  it  out  from 
the  Almanac  for  the  given  Greenwich  mean  time. 

The  Nautical  Almanac  gives  the  semi-diameter,  also  the 
horizontal  parallax,  of  the  sun,  moon,  and  planets. 

To  find  the  apparent  semi-diameter  as  viewed  from  the  ob- 
server's position  on  the  surface : 

From  the  right-angled  triangle  AB'M  (Fig.  105)  sin  S'  =  ~  , 

also,  from  AMC,  f  =  5BJL  =  5™*     and  d'  =  d  °°B  * 
d       sin  zf      cos  h  cos  h 

Substituting  value  of  dr  in  expression  for  sin  8'9 

we  have,  sin  8'  —  ~  COS  ^  ;  but  ~  =  sin  8, 
d  cosh  '         d 

i  t 

therefore,  sin  8'  =  sin  8- 


424  NAUTICAL  ASTRONOMY 

Now  S'  and  8  are  small  angles  and  proportional  to  their 

sines  ;  therefore  8'  =  8  cos  f  (  152  ) 

cos  h 

Prom  this  formula  Sf  may  be  found  when  8,  h',  and  h  are 
known.  As  h  is  greater  than  h',  cos  h  is  less  than  cos  h'  ; 
therefore  S'  is  greater  than  8t  or  the  semi-diameter  increases 
with  the  altitude  of  the  body.  This  excess  is  called  the  aug- 
mentation, but  is  of  appreciable  value  only  in  the  case  of  the 
moon,  for  which  body  it  is  tabulated  in  Table  18,  Bowditch. 

210.  To  find  the  augmentation  of  the  moon's  semi-diameter. 

Let  A$  be  the  augmentation, 

'  cos  ^'  <     cos  ^'  —  cos  ^ 


cos  h'  —  cos  h  =  —  2  sin  £  (h'  —  h)  sin  £  (h'  +  h) 

—  2  sin  \  (h  —  h')  sin  i  (hf 
sin  i  (ft  —  7O  sln  2  (^ 


cos  /& 
Now,  since  &  —  h'  —  p  =  parallax  in  altitude  and  is  very 

small,  2  sin  $(h  —  h')  =  2  sin£=  p  sin  1"  =  P  cos  h'  sin  1". 

^ 

As  AS  is  small,  |  (^  +  Ti)  may  be  taken  as  h'  and  cos  &' 
may  be  substituted  for  cos  h; 

.        .  0       0  (P  cos  ft'  sin  1")  sin  K' 
therefore  AS  =  8  1  _  -  - 

cos  h 

&8  =  8P  sin  h'  sin  \"  =~  82  sin  h'  sin  1"; 

T 
•p 

but  —  sin  1"  is  a  constant  for  any  one  body,  and  may  be  rep- 

resented by  K, 

therefore,  A#  =  K8?  sin  h'. 

In  the  case  of  the  moon,  ^-  =  3.6646  and  log  K  —  5.2496. 

A  more  rigorous  formula  may  be  found  in  Chauvenet's 
Astronomy,  but  the  above  will  not  involve  an  error  greater 
than  -". 


CORRECTING  ALTITUDES  425 

211.  The  following  symbols  may  be  used: 

Inst.  m   (or  _Q.)  Instrumental  altitude  of  sun's  lower  limb. 
Obs.   .Q.  The  above  corrected  for  I.  C. 
Inst.  T-7  Instrumental  altitude  of  sun's  upper  limb. 
Obs.    7!7  The  above  corrected  for  I.  C. 

-0-  True  altitude  of  sun's  center. 
2  O   Twice  the  altitude  of  sun's  lower  limb. 

IT  Altitude  of  moon's  upper  limb. 

Jt,  Altitude  of  moon's  lower  limb. 

-€-  Altitude  of  moon's  center. 
Alt.   *   Altitude  of  star. 

212.  Theoretical  and  practical  methods. — The  various  cor- 
rections to  an  instrumental  altitude  are  applied  as  indicated 
below  in  the  examples  given.     Sometimes  the  theoretical  and 
practical  methods  may  give  slightly  differing  results,  which 
is  a  matter  of  no  importance  at  sea.     Bowditch's  Tables  are 
used.     Tables  II  and  III  in  the  back  of  this  book  were  not 
used  in  the  examples  given  as  illustrations,  as  the  examples 
were  solved  before  their  construction. 

Ex.  149. — January  3,  a.  m.,  1905,  the  instrumental  alti- 
tude of  the  sun's  lower  limb  was  23°  42'  00"  I.  C.  +  1'  20". 
Height  of  eye  45  feet.  Find  the  true  altitude  of  the  center 
of  the  sun. 


THEOBETICALLY. 

PRACTICALLY,  AT  SEA. 

o      /      // 

o     r      ft 

/     // 

Instrumental  0             23  42  00 
I.  C.                             -f         l  20 

Corr. 

23  42  00 

+          8  58 

S.  D.  -f 
I.  C.  -f 

16  18 
1  20 

Observed  0                    23  43  20 
Dip  (Tab.  XIV)          -         6  36 

-e- 

23  50  58 

D.  - 

p.  &  R.  - 

6  36 
2  04 

Apparent  0                   23  36  44 

Corr.  -I- 

8  58 

p.  &  R.  (Tab.  XXB)  -         2  04 

True  0                           23  34  40 

S.  D.                            +       16  18 

True-0-                         23  50  58 

426 


NAUTICAL  ASTRONOMY 


Ex.  150. — January  19,  1905,  the  instrumental  altitude  of 
star  Arcturus  was  29°  22'  10".  I.  C.  +  2'.  Height  of  eye 
45  feet.  Find  the  true  altitude. 


THEORETICALLY. 

PRACTICALLY. 

Oil! 

o      /      // 

i    it 

inst.  alt.  * 

29  22  10 

Inst.  alt. 

*  29  22  10 

I.  C. 

+  2  00 

I.  C. 

Observed  alt.  * 
Dip  (Tab.  XIV) 

+         2  00 

Corr. 
True  alt. 

—         6  19 

D. 
R. 

Corr. 

-  6  36 
-  1  43 

29  24  10 
-         6  36 

*  29  15  51 

—  6  19 

Apparent  alt.  * 
Ref. 

True  alt.  * 

29  17  34 
—         1  44 

29  15  50 

Ex.  151. — January  8,  1905,  the  altitude  of  the  lower  limb 
of  the  sun,  as  observed  with  an  artificial  horizon,  was  34°  36'. 
I.  C.  —  2'.  Eequired  the  true  altitude. 


THEORETICALLY, 
o      /       // 

Instrumental  20         34  36  00 
I.  C.                            -         2  00 

PRACTICALLY. 

20           34  36  00        S.  D. 
I.  C.       -        2  00         p.  &  R. 

/  /; 
+  16  18 
-  2  57 

Observed  20                 34  34  00 
Observed©"                 17  17  00 
p.  &  R.  (Tab.  XXB)  -         2  57 

2)34  34  00 

Corr. 

-f  13  21 

Obs.  ©      17  17  00 
Corr.  ""  +       13  21 

-0-             17  80  21 

True  0                            17  14  03 
S.  D.                             +       16  18 

True  -0-                         17  80  21 

To  correct  an  instrumental  altitude  of  the  moon. — Owing 
to  the  rapid  change  of  the  moon's  semi-diameter  and  hori- 
zontal parallax,  they  must  be  reduced  for  the  Greenwich  mean 
time  of  observation;  also,  as  the  moon  is  nearer  the  observer 
at  all  altitudes  than  when  in  the  horizon,  the  semi-diameter 
must  be  corrected  for  augmentation  (Table  18,  Bowditch). 
The  combined  correction  for  parallax  and  refraction  will  be 
found  in  Table  24,  Bowditch,  in  which  the  arguments  are  the 
horizontal  parallax  at  the  top  and  apparent  altitude  of  the 
moon's  center  at  the  side,  the  parallax  being  at  intervals  of 


CORRECTING  ALTITUDES  427 

one  minute,  the  apparent  altitude  at  intervals  of  ten  minutes 
of  arc.  For  seconds  of  parallax,  enter  the  table  abreast  the 
approximate  correction  where  the  arguments  are  tens  of  sec- 
onds at  the  side  and  units  at  the  top ;  opposite  the  former  and 
under  the  latter  is  the  correction  to  be  added.  The  addi- 
tional correction  for  minutes  of  altitude  will  be  found  in  a 
table  on  extreme  right  of  page,  to  be  applied  as  there  di- 
rected. Hence  the  rules : 

(1)  For  the  given  instant  find  the  corresponding  G.  M.  T. 
for  which  correct  the  moon's  semi-diameter  and  horizontal 
parallax;  also  find  from  Table  18  the  augmentation  of  S.  D. 

(2)  To  the  instrumental  altitude  apply  the  first  correction, 
consisting  of  the  algebraic  sum  of  the  I.  C.,  dip,  and  aug- 
mented semi-diameter.     The  result  will  be  the  apparent  alti- 
tude of  the  moon's  center. 

(3)  With  the  horizontal  parallax  and  the  apparent  alti- 
tude find  from  Table  24,  Bowditch,  the  second  correction  (for 
parallax  and  refraction  combined)  which,  added  to  the  ap- 
parent altitude  of  the  center,  will  give  the  true  altitude  of 
the  center. 

Ex.  152. — January  19,  1905,  in  .longitude  100°  30'  W.,  at 
llh  09m  41s  local  mean  time,  the  instrumental  altitude  of 
the  moon's  lower  limb  was  61°  04'  52"  bearing  N.  I.  C.  +0'. 
Height  of  eye  30  feet  above  the  sea  level.  Kequired  the  true 
altitude. 

L.  M.  T.  Jan.  19  11  09  41  Instrumental  alt._C_61  04  62  S.  D.  +  16  03.6  H.  P. 
Longitude  W  +  6  42  00  1st  correction"  +  10  56.2  Aug.  +  14.6  , 

G.  M.  T.  Jan.  19      17  51  41  Am>flrent  fllt.        -   SliT^  L\a    +    _°°°     58  80.6 


h  m  s 
11  0941 
+    6  4200 

0      1        II 

Instrumental  alt._C_61  04  62 
1st  correction*'          +  10  56.2 

S.  D.  +  16  03.6 
Aug.  +       14.6 

i.e.  +  000 

Dip.   -   628 
1st     I 
corr./+10  68.2 

17  51  41 

17h  51m.68 

Apparent  alt.      -e-6l  15  48.2 
Par.  &  Ref  .  (Tab.  24)  +  27  46 

True  Alt.               >6-  61  43  34.2 

To  be  strictly  accurate  the  H.  P.  found  from  the  Nautical 
Almanac  should  be  further  corrected  for  the  latitude  of  the 
place  of  observation  (Table  19,  Bowditch).  However,  it  is 
usually  disregarded,  as  in  the  above  example. 


428 


NAUTICAL  ASTRONOMY 


Correction  of  a  planet's  altitude. — Theoretically  a  planet's 
altitude  should  be  corrected  for  I.  C.,  dip,  refraction,  parallax, 
and  S.  D. ;  practically,  at  sea,  only  the  first  three  are  applied. 

The  parallax  is  gotten  from  Table  17,  Bowditch,  the  argu- 
ments being  altitude  and  horizontal  parallax. 

Ex.  158. — April  3,  1905,  observed  the  lower  limb  of  planet 
Mars  32°  15'.  I.  C.  +  1'.  Height  of  eye  20  feet.  Kequired 
the  true  altitude. 

THEORETICALLY. 


Instrumental  alt.  of  Mars 
Index  correction 

Obs.  alt.  of  lower  limb 
Dip 

Apparent  alt.  of  lower  limb 
H.  P.  12". 4: 
Par    +      10'M 
Ref.  —  1'  32"/ 

True  alt.  of  lower  limb 
Semi-diameter 

True  alt.  of  center 


The  only  difference  between  the  use  of  planets  and  fixed 
stars  for  navigational  purposes  at  sea  lies  in  the  fact  that 
the  E.  A.  and  declination  of  a  planet  must  be  corrected  for 
G.  M.  T.,  the  corrections  of  the  altitudes  being  practically 
the  same,  I.  C.,  dip,  and  refraction. 

213.  Correction  of  altitude  for  run. — A  ship  at  sea  seldom 
remains  stationary  between  observations;  as  the  ship  moves, 
the  zenith  of  the  observer  describes  an  arc  on  the  celestial 
sphere,  and  the  number  of  minutes  in  the  arc  are  equal  to  the 
number  of  nautical  miles  run  by  the  ship.  A  heavenly  body, 
if  observed  simultaneously  from  the  two  ends  of  the  ship's 
run,  would  have  the  two  zenith  distances  and  hence  the  two 
altitudes  differing  by  an  amount  called,  "the  correction  for 
the  run."  This  must  be  found  whenever  it  is  desired  to  re- 
duce an  observed  altitude  for  a  change  of  the  observer's  posi- 
tion. 


PRACTICALLY, 

AS   AT   SEA. 

O        I        II 

o      r      n 

/    // 

32  15  00 
1  00 

$     32  15  00 
Corr.  —  4  55 

i.e. 

Dip 
Ref. 

Corr 

+  1  00 
-4  23 
-1  32 

32  16  00 
4  23 

(-J1     32  10  05 

-4  55 

32  11  37 

1  22 

32  10  15 

7 

32  10  22 

CORRECTION  FOR  KUN 


429 


Now  suppose  an  observer  whose  zenith  is  Z,  observes  a 
heavenly  body  M;  after  running  a  certain  distance  ZZ1  (Zf 
being  in  one  of  several  different  positions  Z\ ,  Z'2  .  .  .  Z'6), 
he  again  observes  the  same  body.  To  compare  the  two  obser- 
vations, one  must  be  reduced  to  what  it  would  have  been  if 
observed  at  the  other  place. 

Let  M  be  the  position  of  the  heavenly  body,  at  the  first 
observation,  supposed  fixed  during  the  run,  or  as  the  observer's 


FIG.  106. 


zenith  shifts  from  Z  to  Z* ';  ZZ',  the  distance  sailed  in  sea 
miles;  C  =  NZZ',  the  course  sailed  estimated  from  the  ele- 
vated pole ;  and  Z  =  NZM,  the  body's  azimuth. 

If  the  course  is  directly  towards  the  body,  Z  shifts  to  Z\ , 
and  the  body's  zenith  distance  is  lessened,  or  altitude  in- 
creased by  a  number  of  minutes  of  arc  equal  to  the  run  ZZ\ , 
in  sea  miles.  If  the  course  is  directly  away  from  the  body, 
Z  shifts  to  Z'2  and  the  altitude  is  diminished. 


430  NAUTICAL  ASTRONOMY 

If  the  zenith  shifts  to  any  other  position,  by  regarding  tlio 
triangle  ZZ'M  as  a  spherical  triangle  (which  it  is),  the  re- 
duction may  be  obtained  by  a  rigorous  formula  from  Chau- 
venet : 

Afc  =  d  cos  (C— Z)—  4  d2  sin  1'  tan  h  sin2  (C—Z).  (153) 

Now,  with  M  as  a  center,  and  radius  MZ',  describe  an  arc 
cutting  the  first  bearing  line  in  d,  that  is  from  Z'3  in  d3 , 
from  Z\  in  d4 ,  from  Z\  in  <ZB .  Also  drop  perpendiculars 
from  Z'g ,  Z\ ,  and  Z'5  on  the  first  bearing  line  at  ms ,  ra4 , 
%• 

When  the  course  is  away  from  the  body,  the  zenith  going  to 
Z\y  the  correction  is  subtractive  to  the  first  altitude  and 
equals  Zm±  +  m±d±  which  are  respectively  the  first  and  sec- 
ond terms  in  (153). 

When  the  course  is  generally  towards  the  body,  Z  going  to 
Z'B ,  the  correction  is  additive  to  the  first  altitude  and  equals 
Zm$  —  m5d5 ,  the  first  and  second  terms  in  (153). 

When  the  course  is  at  right  angles  to  the  first  bearing  of 
the  body,  Z  going  to  Z'B ,  the  correction  is  —  Zd5 ,  the  second 
term  of  (153) ;  the  first  term  of  (153)  reducing  to  zero. 

Since  the  distance  ZZ'  is  small,  the  second  term  of  (153) 
may  be  neglected,  the  triangles  m±ZZ'±  and  m^ZZ'^  may  be 
regarded  as  plane  triangles,  and  A/i  as  Zm± ,  Zm5 ,  etc. 

Now  Zm±  —  ZZ\  cos  (C  —  (180°  —  Z)), 
or,  ^h  —  d  cos  (C  —  Z). 

Zm5  =  ZZ'6  cos  (Z  —  (7), 
or,  LJi  —  d  cos  (C  —  Z). 

And  A&  is  +  if  (C  ~  Z)  is  <90°, 
Aft  is  — if  (0~Z)  is  >90  , 
Afe  is  0  if  (C  ~  Z)  is  90  . 

Since  A/i  is  zero  when  (C  ~  Z)  equals  90°,  or  is  smaller 
as  (C  ^  Z)  approaches  90°,  it  is  better  to  reduce  that  alti- 
tude for  which  the  difference  of  course  and  azimuth  is 
nearer  90°. 


(154) 


CORRECTION  FOR  RUN  431 

If  the  second  altitude  is  to  be  reduced,  then  C  in  the 
formula  is  the  course  reversed. 

If  a  single  course  and  distance  are  run,  (C  ~  Z)  may  be 
by  compass,  but  if  a  traverse  is  made  from  Z  to  Zr  then  the 
magnetic  (or  true)  course  and  bearing  should  be  used. 

The  traverse  table  may  be  used  by  taking  (C  ~  Z),  or  180 
—  (C  ~  Z)  if  (C  ~  Z)  is  >90°,  for  the  course,  the  run  as 
a  distance,  and  looking  for  the  correction  in  the  difference  of 
latitude  column.  This,  in  minutes  and  tenths  of  a  minute  of 
arc,  is  to  be  added  to  the  first  altitude  if  (C  ~  Z)  is  <90°, 
subtracted  if  (C  ~  Z)  is  >90°. 

Rules  in  finding  correction  for  run. — 

(1)  Take  a  bearing  of  the  heavenly  body  at  each  observa- 
tion. 

(2)  For  the  elapsed  time  between  observations  find  the 
course  made  good  and  distance  run. 

(3)  For  the  distance  in  sea  miles  find  the  altitude  cor- 
rection Aft,  in  minutes  and  decimals  of  a  minute  of  arc,  to 
be  applied  as  already  explained. 

Ex.  154.— On  January  18,  1905,  in  1ST.  Lat.  and  W.  Long., 
a.  m.  time,  the  sextant  altitude  of  sun's  lower  limb,  bearing 
K  137°  E.  (true),  was  22°  18'  20".  I.  C.  +  2'  30".  Height 
of  eye  21  feet.  After  running  NE..  (true)  33  miles,  observed 
a  p.m.  altitude  of  sun's  lower  limb  21°  28'  50".  I.  C.  +  2' 
10".  Height  of  eye  21  feet.  Reduce  first  altitude  to  what 
it  would  have  been  had  it  been  observed  at  the  same  time  at 
the  second  place. 

Course  N  45°  E 

Azimuth      N  137°  E 

o     tt  f 

(C~Z)         =92°.. cos— 8.54282      At  1st  place      O  22  18  20 

d  =  33' log      1.51851       Aft  109 

Ah  =  —  V  09"  =  —  I'.iS      'log  —0.06133      At  2d  place       Q~li2  17  11 


CHAPTER  XVI. 
SOLUTION  OF  THE  ASTRONOMICAL  TRIANGLE. 

214.  In  a  consideration  of  the  astronomical  triangle,   or 
any  of  the  parts  thereof,  in  this  and  succeeding  chapters,  the 
following  notation  will  be  used : 

L  =  latitude. 

hs  =  the  sextant  altitude  of  the  heavenly  body  observed. 

h  =  the  true  altitude  of  the  heavenly  body. 
hr  =  the  apparent  altitude  of  the  body. 
h0  r=  the  meridian  altitude  of  the  body. 

z  =  the  true  zenith  distance  of  the  body. 

z'  =  the  apparent  zenith  distance  of  the  body. 
Z0  =  the  meridian  zenith  distance  of  the  body. 

d  =  its  declination. 

p  =  its  polar  distance  =  90°  —  d  (algebraically). 

Z  =  its  azimuth  measured  from  the  elevated  pole  towards 

the  East  or  West,  from  0°  to  180°. 

Z*$.  =  its  azimuth  measured  from  North  around  to  the  right, 
from  0°  to  360°. 

i  =  its  hour  angle. 
'A  =  its  amplitude. 
M  =  its  position  angle. 

215.  Many  of  the  various  problems  that  confront  a  naviga- 
tor are  solved  by  a  solution  of  the  astronomical  triangle  whose 
sides  are  90°  —  Jif  90°  —  L,  and  90°  —  d  or  p,  and  whose 
angles  are  t,  Z,  and  M. 

Fig.  107  represents  this  triangle  projected  on  the  plane 
of  the  horizon.  By  spherical  trigonometry,  when  any  three 
of  the  parts  are  known,  the  others  can  be  found.  The  posi- 
tion angle  M  is  of  no  importance  to  the  navigator,  and  it  will 
not  be  further  considered.  So  that  for  practical  purposes 


FINDING  TRUE  ALTITUDE 


433 


there  are  given  three  to  find  the  remaining  two  of  the  quanti- 
ties hf  Z,  t,  L,  and  d.  As  the  latter  quantity  is  tabulated 
in  the  Nautical  Almanac,  the  methods  of  finding  the  other 
quantities  will  be  considered  in  the  order  given  above. 

216.  The  parts  of  the  triangle,  when  used  in  the  solutions 
as  given  data,  are  thus  found. 

The  latitudes  and  longitudes  of  places  on  shore,  at  which 
observations  may  be  taken,  are  found  from  charts,  tables  of 

maritime  positions  as  Appendix 
IV,  Bowditch,  or  from  sailing  di- 
rections. At  sea,  they  are  found 
from  previous  determinations 
brought  forward,  or  from  subse- 
quent determinations  carried 
back  by  dead  reckoning;  or  from 
practically  simultaneous  observa- 
tions where  one  body  is  so  favor- 
ably located  that  an  error  in  time 
will  not  affect  the  resulting  lati- 
tude to  be  used  in  a  sight  for 
time  taken  simultaneously,  or  where  one  body  is  near  the 
prime  vertical  and  an  error  in  the  latitude  will  not  affect  the 
resulting  longitude  to  be  used  in  a  sight  taken  simultaneously 
for  latitude. 

Having  on  board  a  chronometer  regulated  to  Greenwich 
mean  time,  and  having  a  comparison  of  the  deck  watch  with 
the  chronometer,  and  the  watch  time  of  observation  of  a  given 
heavenly  body,  the  G.  M.  T.  of  this  observation  is  found  and 
for  it  the  body's  declination  is  taken  from  the  Nautical 
Almanac;  the  polar  distance,  which  equals  90°  —  d  algebraic- 
ally, will  be  less  than  90°,  when  the  name  of  the  declination  is 
the  same  as  that  of  the  latitude;  greater  than  90°,  or 
90°  +  d,  when  its  name  is  different  from  that  of  the  latitude. 


FIG.  107. 


434  NAUTICAL  ASTRONOMY 

The  instrumental  altitude  of  the  heavenly  body  is  reduced 
to  a  true  altitude  (Art.  212),  and  if  the  hour  angle  is  to 
be  one  of  the  given  parts,  it  can  be  found  by  the  methods  of 
Art.  199  when  the  Greenwich  mean  time  and  longitude  are 
both  known. 

The  True  Altitude. 

217.  To  find  the  true  altitude  of  a  heavenly  body  at  a 
given  time  and  place,  when  its  azimuth  is  not  required. 

Here  the  latitude  and  longitude  are  given;  the  Greenwich 
time  is  found  from  the  local  time  and  the  longitude,  and  for 
this  Greenwich  time  the  body's  declination  is  taken  from  the 
Nautical  Almanac.  In  the  case  of  the  sun,  the  declination 
may  be  found  from  page  I  or  page  II  of  the  Almanac  depend- 
ing on  whether  the  Greenwich  time  is  apparent  or  mean  time. 

The  hour  angle  t  of  the  body  is  next  found.  In  the  case 
of  the  sun  West  of  the  meridian,  the  H.  A.  will  be  the  local 
apparent  time;  when  East  of  the  meridian,  the  H.  A.  will  be 
24  hours  —  the  local  apparent  time  considered  astronomically. 

When  the  body  is  the  moon,  a  planet,  or  a  star,  it  will  be 
necessary  to  find  in  order  the  L.  S.  T.,  the  body's  right  ascen- 
sion, and  then  its  hour  angle  (Arts.  192  and  199). 

Applying  to  the  triangle  of  Fig.  107,  the  fundamental  for- 
mula from  spherical  trigonometry 

cos  a  =  cos  &  cos  c  +  sin  &  sin  c  cos  A, 

we  have  sin  h  =  cos(L^d)  —  2  cos  L  cos  d  sin2  %t,  (155) 

or  cos  z  =  cos(L^d)  —  2  cos  L  cos  d  sin2  %t.  (156) 

versinez        1  —  cos  #         .  ,        /..-/.x      M1 
As  haversme  x  =  —  ~  --  =  -  ^  ---  =  sin2  \x  (156)  will 

become  by  substitution,  etc., 

haver  z  —  haver  (L~d)  +  cos  L  cos  d  haver  t. 


Defining  6  by  haver  6  =cos  L  cos  d  haver  tf 

We  have  haver  z  =  haver  (L~d)  +  haver  0 


"1 
.  j 


NOTE.—  When  L  and  d  are  of  different  names  (L~d)  becomes  numerically 
(L  +  d).  The  hour  angle  t  is  usually  given  in  units  of  time,  whereas  the  above 
formulae  require  t  in  arc. 


FINDING  TRUE  ALTITUDE 


435 


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436 


NAUTICAL  ASTRONOMY 


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TIME  AZIMUTHS 


437 


Table  44,  Bowditch,  has  opposite  t  in  the  p.  m.  column  the 
log  sin-J  t  in  the  swie  column;  so,  if  using  formula  (155)  or 
(156),  look  for  t,  expressed  in  time,  in  the  p.  m.  column,  and 
from  the  sine  column,  directly  abreast,  take  out  the  log  sin  ^  t 
which,  when  multiplied  by  2,  will  give  log  sin2  J  t. 

Formulae  (156)  and  (157)  are  useful  in  connection  with  the 
methods  of  "  The  New  Navigation  "  (Art.  308)  . 

Method  of  Time  Azimuths. 

218.  To  find  both  the  altitude  and  azimuth  of  a  certain 
heavenly  body  at  a  given  time  and  place,  or  given  t,  L,  and 
d,  to  find  h  and  Z. 

For  the  given  time  find  the  body's  decimation  (Art.  185), 
and  its  hour  angle  (Art.  199).  Then  in  the  spherical  tri- 
angle PZM  (Fig.  107),  the  following  are  given: 


ZPM  =  tf 
and  it  is  required  to  find 

ZM  =  90°  —  h  and  PZM  =  Z. 

Let  fall  Mm  perpendicular  to  PZ,  call  Pm,  <£;  then  Zm 
—  90°  —  (L  +  <j>),  and  by  Napier's  rules, 

tan  <£  =  cot  d  cos  t  (1)" 

sin  h  =  sin  (L  +  <f>)  sin  d  sec  <j>        (2) 
cotZ   =  cot  t  cos  (L  +  <#>)  cosec  <f>  (3)^ 
Following  Chauvenet's  methods,  the  above  can  be  put  into 
a  more  convenient  form.     If  <f>  =  90°  —  <£',  the  above  become 
tan  <£'  =  tan  d  sec  t  (V) 

cos     '-£   sin 


(158) 


sn       = 


sin 
cot  t  sin 


0080' 


(2) 
(3) 


(159) 


In  (159),  </>'  is  taken  out  in  the  same  quadrant  as  i  and  is 
given  the  same  sign  as  the  declination ;  that  is,  if  the  declina- 
tion is  of  the  same  name  as  the  latitude,  it  is  +,  and  $'  is 


438  NAUTICAL  ASTRONOMY 

marked  + ;  if  the  declination  is  of  a  different  name  from  the 
latitude,  it  is  ( — ),  and  </>'  is  marked  ( — ).  The  mere  fact 
of  t  being  E.  or  W.  has  no  influence  on  the  signs  of  the 
functions  sec  t  and  cot  t.  If  t  is  E.  or  ( — ),  the  body  is 
East  of  the  meridian  and  the  azimuth  is  marked  East ;  if  Hs 
W.  or  +>  the  body  is  West  of  the  meridian,  and  the  azimuth 
is  marked  West ;  in  other  words,  the  azimuth,  being  restricted 
to  180°,  is  reckoned  from  the  elevated  pole  (or  the  North 
point  of  horizon  in  North  latitude,  the  South  point  in  South 
latitude)  towards  the  East  or  West  according  as  the  body  is 
East  or  West  of  the  meridian  as  indicated  by  the  hour  angle. 
Again,  for  emphasis,  let  it  be  repeated  that  a  t's  mark  E.  or 
W.  does  not  affect  the  sign  of  its  function  in  the  above  for- 
mulae. However,  the  signs  of  functions  of  other  quantities 
must  be  followed,  and  care  must  be  exercised  to  do  so. 

When  t  =  6  hours,  (ft'  =  90°,  and  the  formula  for  Z  (159) 
becomes  of  an  indeterminate  form.  However, 

cos  t  =te**T  and  cot  t  =       *|5-* 

tan  (ft  tan  <ft  sin  t 

and  by  substitution  in  (159) 

cot  Z  =  -m  \ft  ~    /    ^n —  (160) 

sin  <ft  sm  t 

which  can  be  used  when  t  is  near  6  hours. 

Formulas  (158)  may  be  simplified  in  the  case  of  Polaris 
(a  Ursae  Minoris)  because  of  its  small  polar  distance. 

Since  tan  (ft  =  tan  p  cos  t  (<ft  and  p  being  very  small), 
<ft  =:  p  cos  t. 


sin  ft  =  j     Qr  approximately  ft  =  L  +  <£. 

COS  <J) 


cos 


sn  (f)  cos      +  $         cos 

but  (f>  is  so  small,  cos  <£  is  near  unity;  therefore, 

Z  =  p  sin  t  sec  (L  +  <£)  approximately. 
Example  157,  and  Ex.  158  worked  first  for  North  latitude, 
then  for  South  latitude,  will  illustrate  the  method  (formulae 
159). 


A  TIME  AZIMUTH  OF  SUN 


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TIME  AZIMUTHS  OF  SUN  441 

Now  work  the  above  example  for  Lat.  28°  30'  S. 

o     t    n 

d            =  — 19  40  28.8                tan  —  9.55334    sin    —    9.52722 

t             =  E.  56  08  37.6                sec       10.26406                                      cot       9.82664 

$'          =  (— )  32  41  32                  tan  —  9.80740    cosec—  10.26750       sec     10.07490 

L           =  2830         S 

<J>'  —  L  =  (— )     61  11  32                                               cos  +       9.68293       sin  —  9.94263 

h  =  172846  sin+       9.47765  

Z           -  S 124  56  05    B  (ZN=55°  03'  55")                                          cot  —  9.84417 

Here  latitude  is  S.  and  +>  declination  is  N.  and  ( — ) ; 
signs  should  be  followed  as  indicated.  The  azimuth  is  reck- 
oned from  the  elevated  or  S.  pole,  and  to  eastward. 

The  above  formula  for  azimuth  is  of  great  importance  in 
finding  the  deviation  of  the  compass :  Suppose  that  the  navi- 
gator about  8h  08m  45s  a.  m.,  January  3,  1905,  in  Lat. 
7°  28'  06"  N.,  Long.  150°  09'  54"  W.,  had  observed  the  bear- 
ing of  the  sun  per  compass  to  be  Z^  =  108 -J°,  ship's  head 
45°  (p.  c.),  and  it  is  required  to  find  the  deviation  (see 
Ex.  157).  Working  the  time  azimuth  we  have 

Sun's  bearing  (true)  119°  04'  40" 

Sun's  bearing  (p.  c.)  108    30 

Compass  error  =  +10°  34'  40" 

Variation  =  +  8 

For  45°  (p.  c.),  deviation  =  +    2°  34'  40" 

Time  azimuths  may  be  obtained  as  above  by  the  solution 
of  the  astronomical  triangle,  from  azimuth  tables,  or  by 
graphic  methods  from  azimuth  diagrams. 

Instead  of  deducing  the  above  formula  for  Z  by  Napier's 
rules,  the  third  and  fourth  of  Napier's  analogies  may  be  used. 
These,  applied  to  the  astronomical  triangle,  give 

tan  \  (Z  +  M )  =  cot  It  cos  J  (L  —  d)  cosec  £  (L+d)  J 

The  azimuth  Z,  however  found,  should  be  expressed  for 
practical  purposes  in  the  form  of  Z^  which  is  measured  from 
North,  around  to  the  right,  from  0°  to  360°. 


442  NAUTICAL  ASTRONOMY 

The  Altitude-Azimuth  Method. 

219.  To  find  the  azimuth  of  a  heavenly  body  from  its  ob- 
served altitude  at  a  given  place. 

Noting  the  time  of  observation  by  a  watch  compared  with 
a  chronometer  regulated  to  Greenwich  mean  time,  the  G-.  M.  T. 
of  observation  is  found,  and  for  this  the  declination  of  the 
body  is  taken  from  the  Nautical  Almanac.  Knowing  the 
latitude,  and  reducing  the  sextant  altitude  to  a  true  altitude, 
the  three  sides  of  the  astronomical  triangle  are  known. 

By  spherical  trigonometry, 

cos2  *A=  *™  8  Bin  (8  -a) 
sin  b  sin  c 

in  which  a,  b>  and  c  represent  the  three  sides  of  the  triangle, 


Applying  this  formula  to  the  astronomical  triangle  PZM, 
(Fig.  107), 

A  =  Z  =  the  azimuth  of  the  heavenly  body  ; 
a  =  p  =  the  polar  distance  of  body  ; 
I  =  90°  —  L  —  the  co.  latitude; 
c  =  90°  —  h  =  the  co.  altitude  of  the  body. 

o__#  +  5  4-  c_  Qno      L  +  h  —  p 

-~~  ~- 


Therefore,  cos*  J  Z  =  cos      cos 

Now  letting  s  =  %  (L  +  h  +  p), 

then  \  (L  +  h  —  p)  =  s  —  p, 


and  cos  1  Z  =      coss  cos  (g~ 

? 


cos  L  cos  h L-      (162) 

=  \/  cos  s  cos  (s  — p}  sec  L  sec  h.  } 


ALTITUDE-AZIMUTH  443 


We  also  have  from  spherical  trigonometry 

orf  I  A  =  dn(fl-8)gin(fl-c)   where 
sin  b  sm  c 

In  triangle  PZM  (Fig.  107), 

a  =  90°  —  d, 
b  =  co.  L, 
c  =  90°  —  h. 
fl  +  I  +  c          0      h  +  d  —  co  L 


cos 

Therefore,  sin2  J  Z  = 


cos  L  cos  h 

Now  letting  s'  =  %  (h  +  d  +  co.  L), 
s'  —  d  =       h      co.L  —  d 


rzz    / 

V 


ssn  s- 


(163) 
cos  I/  cos  £         J 

(162)  is  preferred  when  Z  is  >90°,  (163)  when  Z  is  <90°. 
This  problem  is  known  as  the  altitude-azimuth. 

Formula  (162)  is  more  convenient  for  use  in  connection 
with  the  problem  of  finding  the  hour  angle  of  a  heavenly  body 
and  is  more  generally  used. 

If  the  bearing  of  the  heavenly  body  is  observed  by  compass 
at  the  time  of  getting  its  altitude,  or  if  the  bearing  at  this 
time  is  interpolated  for  from  previous  and  subsequent  bear- 
ings per  compass,  the  error  of  the  compass  can  be  found. 

It  has  already  been  shown  that  compass  error  is  the  differ- 
ence between  the  true  and  compass  bearings  of  a  heavenly 
body  at  the  same  instant,  and  is  marked  E.  when  the  true 
bearing  is  to  the  right  of  the  compass  bearing,  W.  when  the 
true  bearing  is  to  the  left  of  the  compass  bearing. 


444  NAUTICAL  ASTRONOMY 

The  above  formulae  are  adapted  for  use  with  almost  any 
tables,  but  certain  tables,  like  Inman's  Tables,  are  in  use  and 
contain  log  haversines  and  ^  log  haversines  (the  term  haver- 

sine  <£,  meaning  yer^m  ^  or  sin2  4  <f>),  and  the  formula  for 

fO 

azimuth  in  the  following  form  can  be  most  conveniently  used 
by  those  navigators  furnished  with  Inman's  Tables. 

From  trigonometry  sin  J  A  =    /sin(ff—  ft)  sm(ff—  c) 

Y  sin  b  sin  c 


in  which 


8  -  *±*±c>_  90o      h  +  L-p 

2  ~2~     ' 


Therefore,  sin-  JZ  =  -  og      co  s 


haversine  Z  =  '  (h  " 


cos  L  cos  h 

log  hayer  ^=  J  log  haver  (p—(li—L))  +  £  log  haver  (p  -J-  (Ji—  L)*) 
+  log  sec  L  +  log  sec  h.  (164) 

The  solution  by  formula  162  is  illustrated  in  Ex.  159  on 
page  445.  The  points  to  be  noted  in  that  example  are  the  fol- 
lowing :  The  G.  M.  T.  being  >  12  hours,  the  declination  is 
taken  out  for  the  next  date  and  corrected  backwards.  The  de- 
clination being  of  a  different  name  from  the  latitude,  the  polar 
distance  is  >90°.  The  true  azimuth  is  marked  from  the 
South  point  of  horizon  because  latitude  is  South;  and,  East 
as  the  body  is  East  of  the  meridian. 

Amplitudes. 

220.  The  amplitude  of  a  heavenly  body  is  its  angular  dis- 
tance from  the  East  or  West  point  when  in  the  true  horizon, 
and  is  marked  N.  or  S.  according  as  it  is  N".  or  S.  of  that 


ALTITUDE- AZIMUTH  OF  SUN 


445 


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446 


NAUTICAL  ASTRONOMY 


paint.  In  other  words,  it  is  the  complement  of  the  azi- 
muth when  the  body  is  in  the  true  horizon.  In  Fig.  108 
and  Fig.  109,  let  PZM  be  a  projection  of  the  astronomical 
triangle  on  the  plane  of  the  horizon,  the  body  M  being  in  the 
horizon,  and  in  both  cases  just  rising.  Let  NZS  be  the  celes- 
tial meridian,  WQE  the  celestial  equator,  WZE  the  prime 
vertical,  and  EM  =  A  =  amplitude  =  90°  —  NM  =  90°  —  Z. 
In  Fig.  108,  the  latitude  and  declination  are  of  the  same 
name, 

PM  =  90°  —  d,PN  =  Lai,  NM  =  9Q°—A=Z, 
and  in  the  triangle  PNM,  the  angle  PNM  is  a  right  angle. 


FIG.  108. 


In  Fig.  109  the  latitude  and  declination  are  of  a  different 
name;  therefore, 

PM  =  90°  +  df  PN  =  Lai,  NM  =  90°  ^  A  =  Z, 
and,  as  before,  PNM  is  a  right  angle. 

Applying  Napier's  rules  to  the  triangle  PNM, 

cos  PM  =  cos  PN  cos  NM, 
cos  (90°  ±d)=  cos  L  cos  (90°  ±  A)  , 
sin  d  =  cos  L  sin  A  , 
sin  A  =.sin  d  sec  L.  (165) 

It  is  evident  from  the  two  figures  that  a  body  will  rise  and 


TIME  FOR  AN  AMPLITUDE  447 

set  to  the  northward  or  southward  of  the  prime  vertical  ac- 
cording as  its  declination  is  N.  or  S. 

Amplitudes,  computed  by  formula  (165),  are  tabulated  in 
Table  39,  Bowditch,  for  which  the  arguments  are  declination 
at  the  top  and  latitude  in  the  side  column;  the  true  ampli- 
tude is  found  under  the  former  and  opposite  the  latter. 

The  azimuth  tables  give  not  only  the  azimuth,  which  is 
the  complement  of  the  amplitude  when  the  body  is  in  the 
horizon,  but  also  the  times  of  rising  and  setting. 

Time  for  observing  an  amplitude. — This  problem  sup- 
poses the  body  to  be  in  the  true  horizon,  that  is,  the  true  alti- 
tude of  the  center  to  be  0°.  If  h  is  a  true  altitude,  In!  an 
observed  altitude  of  the  center,  then  h  =  hf  —  D  —  R  +  p, 
or  hr  =  h  +  D  +  R  —  p,  but  when  the  sun's  center  is  in  the 
true  horizon  h  =  0°,  D  =  dip  depending  on  height  of  eye, 
R  z=  36'  29",  p  =  9".  Therefore,  as  observed,  the  altitude  of 
the  center  will  be  36'  20"  +  dip  above  the  visible  horizon; 
hence  the  rule,  in  taking  an  amplitude  of  the  sun,  is  to  observe 
the  bearing  per  compass  of  its  center  when  its  center  is  about 
one  sun's  diameter,  or  the  lower  limb  a  semi-diameter,  above 
the  visible  horizon.  Note  at  the  same  instant  the  ship's  head 
(p.  s.  c.),  angle  and  direction  of  heel,  and  the  time  by  a 
watch  compared  with  a  chronometer  regulated  to  G.  M.  T. 
Or,  the  bearing  of  the  center  in  the  visible  horizon  may  be 
obtained  by  taking  the  mean  of  the  bearings  of  the  upper 
and  lower  limb  of  the  sun  when  rising  or  setting,  and  by 
applying  a  correction  for  the  vertical  displacement  from 
Table  40,  Bowditch,  the  observed  amplitude  may  be  reduced 
to  what  it  would  have  been,  if  taken  when  the  sun's  center 
was  in  the  true  horizon. 

Stars  are  not  often  available  for  amplitudes,  except  in  the 
cases  of  very  bright  stars  or  planets  before  setting,  and  then 
the  altitude  should  be  36'  29"  +  dip  above  the  horizon.  If 
observed  in  the  visible  horizon,  the  correction  from  Table  40, 
Bowditch,  must  be  applied.  In  the  cases  of  the  sun,  a  star, 


448  NAUTICAL  ASTRONOMY 

or  a  planet,  this  correction  is  applied  to  the  right  at  rising  in 
North,  or  setting  in  South  latitude;  otherwise  to  the  left. 

The  moon  should  not  be  observed  for  an  amplitude,  be- 
cause when  it  has  its  center  in  the  true  horizon,  the  center  is 
not  visible,  due  to  the  excess  of  the  parallax  over  the  refrac- 
tion. When  the  moon's  center  is  seen  in  the  visible  horizon, 
or  h'  =  0,  the  true  altitude  of  the  center  is  ( -f-  H.  P.  — 
refraction  —  dip) ;  now,  as  the  H.  P.  averages  about  58',  the 
refraction  about  36',  the  dip  being  dependent  on  the  height 
of  the  eye,  when  the  moon's  center  is  just  seen,  rising  or  set- 
ting, on  the  visible  horizon,  it  is  in  reality  22'  —  dip,  or 
about  one  of  its  semi-diameters  above  the  true  horizon ;  for  this 
reason  the  moon  should  not  be  considered  available  for  ampli- 
tudes. 

Amplitudes  of  the  Sun. 

Ex.  160.— At  sea,  in  Lat.  40°  20'  N.,  Long.  60°  15'  W., 
about  6h  20m  p.  m.  local  apparent  time,  on  April  5,  1905, 
the  bearing  per  standard  compass  of  the  sun's  center  at  the 
time  when  the  center  was  estimated  to  be  one  diameter  above 
the  visible  horizon  was  W.  10°  30'  N.  Eequired  the  com- 
pass error. 

It  is  first  necessary  to  find  the  Greenwich  time  and  then  the 
declination.  If  the  approximate  time  had  not  been  given,  it 
could  have  been  found  from  sunset  or  azimuth  tables;  the 
latter,  however,  would  also  give  the  azimuth  =  (90° — the 
amplitude) . 


L.  A.  T. 
A  West 
G.  A.  T. 

or  April  5, 

h 

=    6 
4 

in     s 

20  00 
01 

Dec. 
Corr. 
Dec. 

N 
N 
N 

O       t      ft 

5  55  27.4 
9  50.3 

H.D. 
Corr. 

N 
N 

57".03 
lOh.35 
690".26 

=  10 
IQi 

21  00 
'.35 

6  05  17.7 

By  Computation.  By  Inspection  (Table  39  Bow.) 

L  =  40K°  N 
d  =  6°  1   N 


o    i   n  ,  _  ,n,,0  ^  \  o 

L       =     40  20      N       sec  10.11788    *     \  Tr*   fl  (  True  Amp.  =  W       7.93 


Dec.=       60518N       sin    9.02553  .  _ 


A       =W75947N       sin    9.K341  s  error 

o      /    ;/ 

True  Amplitude  =  W     7  59  47   N  °    '    "  w 

Amplitude  (p.  s.  c.)  W    10  30       N 

Compass  error  2  30  13  W  _ 


AMPLITUDES 


449 


Ex.  161.— At  sea,  in  Lat.  38°  S.,  Long.  85°  E.,  at  sunrise 
(L.  A.  T.  about  4h  46m  a.  m.),  January  9,  1905,  the  bearing 
(p.  s.  c.)  of  the  sun's  center  at  the  time  when  the  center  was 
estimated  to  be  one  diameter  above  the  visible  horizon  was 
E.  20°  S.  Eequired  the  compass  error. 


h      m 

L.  A.  T.  of  sunrise  =    16  46  00    Jan.  8. 
A  East  =      540 

G.  A.  T. 


Dec.  S    22  17  37.1       H.  D.      N  19".86 

Corr.  N         3  40.4       G.  A.  T.        llh.l 

11  06        Jan.^8.       Dec.  S    22  13  56.7       Corr.       N  220".4 


By  Computation. 

L          =    38°          S sec  10.10347 

d          =S22°13'57" sin    9.67791 

A          =E28  41  44  S sin    9.68138 

Q      I     II 

True  Amplitude  E    28  41  44  S 
Ampl.  (p.  s.  c.)     E    20  S 

Compass  error  8  41  44   E 


By  Inspection  (Tab.  39  Bow.) 


TrueA-np. 
Amp.  (p.  s.  c.) 
C.  E.  =  8°  40'  48"  E  = 


E20       S 
8.68  E 


Amplitude  of  a  Star. 

Ex.  162.— At  sea,  in  Lat.  40°  K,  Long.  30°  W.,  the  bear- 
ing (p.  s.  c.)  of  star  Sirius  when  in  the  visible  horizon,  at 
setting,  was  W.  2°  S.  The  star's  declination  was  S.  16°  35; 
20".  Find  compass  error. 


By  Computation. 

L  =  40°  N sec  10.11575 

d  =  S  16°  35'  20" Sin     9.46561 

True  Amplitude  W  21  62  56  S    sin    9.57136 
Observed  Amp.  W  2°    S ) 
Tab.  40  Corr.  left     0.6  S  f  Of* 

Comp.     Amp.  =  W  3°.6  S  =  W  2    36    00    S 
True  Amplitude  =W  21    62    66    S 

Compass  error  19    16  66   W 


By  Inspection. 


o 
21.92    S 


L  40°  N  I  True  Amp. 

d  16.6  S' 

Amp.  (p.  c.)  W    2.6      S 

C.  E.  =    19°  19'  12"  W  =  19.32  W 


Amplitude  of  a  Planet. 

Ex.  163.— On  January  6, 1905,  about  12h  40m  a.  m.  L.M.T., 
in  Lat.  35°  K,  Long.  150°  15'  E.,  Jupiter's  bearing  (p.  s.  c.) 


450  NAUTICAL  ASTRONOMY 

when  in  the  visible  horizon,  at  setting,  was  W.  7°  30'  S.     Re- 
quired the  compass  error. 


L.  M.  T.  of  bearing  Jan.  5 
A.  East                                (-) 
G.  M.  T.  Jan.  5, 

h    m 
12    40 
10    01 

Dec.    N 
Corr.  N 
Dec.    N 

0    /     /( 

701  14 
12.2 

H.  D.      N   4".61 
G.  M.  T.       2h.65 
Corr.      N  12".22 

,2    39 
2h.65 

7  01  26.2 

By  Computation.  By  Inspection. 

L  =  35°  N         .        .        .        .          sec  10.08664       L  =  35°  N  \  True  AmT>   w  ao  A  w 
d  =  7°  01'  26"  N    .  .       .    sin    9.08734       d  =    7°  N  f  1]  ae  Amp<  W 

True  Amplitude  W  8°  35' 05"  N    sin    9.17398       Amp.  (p.  s.  c.)  W  jB S 

C.  E.  =  16°.6  E 

Obs.  Amp.  W  7°  30'  S 

Tab.  40  Corr.  left _30          o    /    „ 

Compass  Amp.=  W  8°S   =  W  8  00  00   S 
True  Amplitude  =  W  8  35  05   N 

Compass  error  16  35  05  E 

Azimuth  Tables. 

221.  The  azimuth  tables  issued  by  the  Navy  Department 
embody  two  separate  publications,  No.  71  and  No.  120. 

In  both,  the  azimuth  is  given  at  intervals  of  ten  minutes 
of  hour  angle,  the  arguments  being  latitude,  declination,  and 
hour  angle.  In  No.  71,  the  latitude  runs  from  0°  to  61° 
at  intervals  of  1°,  the  declination  from  0°  to  23°  at  inter- 
vals of  1°.  No.  71  is  especially  adapted  to  the  case  of  the 
sun,  though  applicable  to  the  cases  of  all  bodies  of  a  declina- 
tion less  than  23°  North  or  South.  The  hour  angle  is 
given  in  the  p.  m.  column,  12  hours  —  H.  A.  in  the  a.  m.  col- 
umn, the  H.  A.  in  case  of  the  sun  being  local  apparent  time. 
When  the  body  is  in  the  true  horizon,  the  hour  angle  is  the 
time  of  sunset  and  12  hours  —  H.  A.  the  time  of  sunrise.  The 
azimuth  is  also  given  when  the  body  is  in  the  true  horizon. 

No.  120  is  intended  for  use  with  the  stars,  planets,  and  the 
moon.  It  is  tabulated  for  latitudes  from  0°  to  70°  and  de- 
clinations from  24°  to  70°  (see  Appendix  C). 

To  use  the  Azimuth  Tables. 

Take  each  argument  to  its  next  lower  tabulated  amount 
and  find  the  azimuth  corresponding  from  the  tables,  placing 


AZIMUTH  TABLES 


451 


it  on  the  first  line  in  each  column  of  the  tabulated  form 
following. 

Now  consider  two  of  the  above  arguments  unchanged  and 
the  third  to  be  of  the  next  higher  denomination  till  each  argu- 
ment has  been  successively  changed;  find  from  the  azimuth 
tables  the  azimuth  corresponding  to  each  set  of  arguments, 
placing  the  result  in  the  column  whose  name  at  the  top  in- 
dicates the  changing  argument,  and  just  below  the  azimuth 
first  taken  out. 

Then,  having  the  change  in  azimuth  for  an  interval  of  one 
argument,  find  the  change  for  the  given  fraction  of  that 
interval.  The  algebraic  sum  of  the  changes  for  fractions  of 
all  intervals  is  a  correction  to  be  applied  by  sign  to  the  azimuth 
first  taken  out.  Having  found  Z,  express  it  in  the  form  of  ZN. 

Though  this  may  be  done  mentally,  the  example  and  form 
below  will  indicate  the  method  of  solution. 

Ex.  164.— In  latitude  30°  30'  N.  find  the  azimuth  of  a 
heavenly  body  whose  declination  is  21°  10'  N.  and  whose 
H.  A.  is  +  4h  13m. 


Arguments. 


Differences  for 


10  Min.  of 
Hour  Angle. 


Lat.  30°  N 
Dec.  21°  N 
H.  A.  +  4h  10™ 


N  83°  28'  W 


82°  23' 


l°of 
Declination. 


N  83°  28'  W 


82*  21' 


l°of 
Latitude. 


N  83°  28'   W 


84°  08' 


Change  for  10""  of  H.  A        —65' 
«         «      1°  ot  Dec. 
«         «       1°  of  Lat. 

Change,  therefore,  for  3m  of  H.  A. 
"  "  «'    10'  of   Dec. 

«  "  «    30'  of   Lat. 


-67' 


+  40' 


Corr.     —  10.7 
N  83°  28'    W 

Required  azimuth- =  N  83°  17'. 3  W  and  ZN  =  276°  42'. 7 


452  NAUTICAL  ASTRONOMY 

222.  In  both  an  altitude  azimuth  and  a  time  azimuth  the 
declination  may  be  regarded  as  accurately  known;  in  the 
former  the  altitude  and  latitude,  in  the  latter  the  time  and 
latitude  are  liable  to  error.  Therefore,  it  is  necessary  to 
consider  the  effect  on  the  resulting  azimuth  of  small  errors  in 
data,  and  the  determination  of  the  most  favorable  position  of 
a  heavenly  body  for  observations  for  Z. 

(1)  In  an  altitude  azimuth  to  find  the  variation  in  Z  due 
to  a  variation  in  h. 

Taking  the  fundamental  trigonometric  formulae 

cos  a  =  cos  &  cos  c  +  sin  &  sin  c  cos  A, 
sin  A  sin  &  =  sin  B  sin  a, 

and  substituting  A  =  Z,  B  —  M,  a  =  90°  —  d,  I  =  90°  —  L, 
c  =  90  —  hf  we  have 

sin  d  =  sin  L  sin  h  +  cos  L  cos  h  cos  Z  "1 
sin  Z  cos  L  =  sin  M  cos  d  L     (160) 

sin  Z  cos  h  =  sin  t  cos  d 

By  differentiation,  h  and  Z  variable, 

0  =  sin  L  cos  lidli  —  cos  L  cos  Z  sin  h dh 
—  cos  L  cos  h  sin  ZdZ 
dZ       sin  L  cos  h  —  cos  L  cos  Z  sin  h  ,       x 

__-  =  . . (  1 D  7  ) 

ah  cos  L  cos  h  sin  Z 

From  trigonometr}',  sin  a  cos  B— cos  &  sin  c — sin  &  cos  c  cos  A. 
By  substituting  the  above  quantities  in  this  formula,  we  have 

cos  d  cos  M  =  sin  L  cos  h  —  cos  L  sin  h  cos  J£. 

Substituting  cos  d  cos  M  and  the  value  of  cos  L  from  (166) 
in  (167),  we  have 

dZ                      cos  d  cos  M  ,  ir        7     /^/>o\ 

^_  -  =  cot  M  sec  h.   (168) 

rtA        sin  3f  cos  d  cosec  Z  cos  h  sin  Z 


EFFECT  ON  Z  OF  ERRORS  IN  h  AND  L  453 

Also,  by  substituting  cos  d  cos  M  and  the  value  of  cos  h  from 
(166)  in  (167),  we  have 

dZ=  --        COB  d  cos  M  c08MsQcLcosecL   (169) 

all      cos  L  sin  £  cos  d  cosec  ^  sin  Js 

(168)  shows  ^f  to  be  least  when  M  is  90°  and  h  =  0°. 

w/£ 

(169)  shows  ^  to  be  least  when  M  is  90°,  L  is  0°  and  i  is 
6  hours. 

(2)  To  find  the  variation  in  Z  due  to  a  variation  in  L.  — 
Differentiating,  sin  d  =  sin  L  sin  h  +  cos  L  cos  h  cos  Z; 
regarding  L  and  Z  as  variables,  we  have 

0  =  sin  /i  cos  LdL  —  cos  7t  cos  Z  sin  LdL 

—  cos  h  cos  L  sin  ZdZ 
dZ     sin  h  cos  L  —  cos  h  sin  Jv  cos  Z 
dZ~~  cos  h  cos  L  sin  Z 

By  trigonometric  substitution  as  in  the  previous  case, 
cos  d  cos  t  =  sin  h  cos  L  —  cos  h  sin  L  cos  2T. 

Substituting  cos  d  cos  £  and  the  value,  of  sin  Z  in  terms  of 
M  from  (166)  in  (170),  we  have 

cos  d  cos  £ 


cos  h  cos  .L  sin  M  cos  d  sec 
cos  t 


—  cos  t  sec  »  cosec  M. 


cos      sin 

Substituting  cos  d  cos  /  and  the  value  of  sin  Z  in  terms  of  t 
from  (166)  in  (171),  we  have 
CIZ  cos  d  cos  t 


dL       cos  /i  cos  L  sin  £  cos  d  sec  h 


=  cottsecL.  (172) 


(171)  shows  ^?  to  be  least  when  M  is  90°,  h  is  0°,  and  t  is 
6  hrs. 

rJ  *7 

(172)  shows  -rr  to  be  least  when  Hs  6  hrs.  and  L  is  0°. 

(I  x/ 


454  NAUTICAL  ASTRONOMY 

In  a  Time  Azimuth. 

(3)  To  find  the  variation  in  Z  due  to  a  variation  in  t.  — 

Taking  the  trigonometric  formula  cot  A  sin  C  =  sin  &  cot  a 
—  cos  &  cos  C,  and  as  in  the  first  and  second  cases  above, 
substituting  A  =  Z,  C  =  t,  a  —  90°  —  d,  &  =  90°  —  L, 
we  have  an  expression  involving  only  those  quantities  used  in 
the  solution  of  a  time  azimuth,  namely, 

cot  Z  sin  i  =  cos  L  tan  d  —  sin  L  cos  t. 
By  differentiation,  Z  and  t  regarded  as  variables, 

—  sin  t  cosec2  ZdZ  -\-  cot  Z  cos  tdt  =  sin  .L  sin  id/, 
cos  Z  cos  i  —  sin  L  sin  i  sin  Z 


dt  sin  i  cosec  Z 

From  trigonometry,  cos  5  =  —  cos  A.  cos  C  -j-  sin  A  sin  C  cos  &. 
By  making  the  same  substitutions  as  before,  we  have 
cos  M  =  —  cos  Z  cos  i  +  sin  Z  sin  i  sin  L, 
or,  cos  If  =  —  (cos  Z  cos  /  —  sin  L  sin  i  sin  Z}  . 

•j  &  ,  ______  ,  pr\Q   lyT 

Therefore,  ~=  -  =  —  cos  M  sin  Z  cosec  t.  (173) 

at     cosec  Z  sin  t 

From  (166),  sin  If  cosec  /  =  cos  d  sec  7i; 

therefore,  -^  =  —  cos  M  cos  d  sec  h.     (174) 
$£ 

(173)  shows  ^?  to  be  least  when  M  is  90°  and  t  =  Q  lira. 

(174)  shows  ^f  to  be  least  when  M  is  90°  and  7t  =  0. 

dt 

Conclusions. 

It  is  thus  seen  that  the  ideal  circumstances  for  observations 
in  the  determination  of  the  azimuth  of  a  heavenly  body,  and 
hence  of  deviation  of  the  compass,  would  be  when  the  ob- 
server is  on  or  near  the  equator,  and  the  heavenly  body  is  on 
the  prime  vertical  in  the  true  horizon,  rising  or  setting,  its 


ASTRONOMICAL  BEARING  455 

position  angle  being  90°.  However,  in  the  determination  of 
the  deviation,  azimuths  can  be  taken  at  any  time,  provided 
the  change  of  azimuth  is  not  too  rapid,  or  the  altitude  so 
great  as  to  make  important  the  errors  arising  from  the  want 
of  verticality  of  the  sight  vanes  of  the  azimuth  circle  or 
pelorus. 

True  Bearing  of  a  Terrestrial  Object. 

223.  In  the  survey  of  a  harbor,  it  is  necessary  to  know  the 
azimuth  of  at  least  one  of  its  lines,  that  is  the  true  bearing  of 
some  one  station  from  another,  that  other  lines  may  be  laid 
off  in  their  proper  directions,  and  a  meridian  line  drawn  upon 
the  chart. 

It  may  often  be  desirable  to  determine  the  true  bearing  of 
a  distant  peak  or  point  in  finding  compass  error. 

If  a  terrestrial  object,  whose  true  bearing  has  been  deter- 
mined from  a  shore  position,  be  observed  from  the  same  posi- 
tion by  compass,  the  difference  between  the  two  bearings  will 
give  the  variation  of  the  locality. 

The  azimuth  of  a  terrestrial  object  may  be  found  by  com- 
bining in  the  proper  way  the  angle  between  the  terrestrial 
object  and  a  heavenly  body  with  the  azimuth  of  the  same 
heavenly  body  determined  at  the  same  instant. 

224.  First  method. — The  angle  between  the  two  objects 
may  be  determined  by  using  the  azimuth  circle  of  the  standard 
compass  or  pelorus  on  board;  but  ashore,  where  more  refined 
observations  would  be  needed,  it  may  be  measured  by  a  theo- 
dolite.    This  instrument  is  brought  to  bear  on  the  heavenly 
body,  and  the  time  is  noted  or  its  altitude  measured  by  a  sec- 
ond observer  simultaneously  with  the  reading  of  the  circle. 
In  the  absence  of  a  second  observer,  the  altitude  of  the  heavenly 
body  may  be  observed  before  and  after  the  circle  is  read ;  and 
from  the  times  noted  and  their  corresponding  altitudes,  by  in- 
terpolation, the  altitude  at  the  instant  of  reading  the  circle 


456 


NAUTICAL  ASTRONOMY 


may  be  obtained.  Turn  the  telescope  in  azimuth,  bringing  it 
to  bear  on  the  terrestrial  object  and  read  the  circle  again. 
The  difference  of  the  two  readings  of  the  circle  will  be  the  dif- 
ference of  azimuths  of  the  two  objects,  which  being  applied  to 
the  true  azimuth  of  the  heavenly  body  found  by  (1)  the  time- 
azimuth  method  or  (2)  the  altitude-azimuth  method  will  give 
the  true  azimuth  of  the  terrestrial  object. 

When  the  heavenly  body  has  an  appreciable  diameter,  as  in 
the  case  of  the  sun,  both  limbs  must  be  observed  thus :   Bring 


FIG.  110. 


FIG.  111. 


the  vertical  wire  of  the  telescope  tangent  to  one  limb  of  the 
sun,  the  neutral  glass  being  used  on  the  eye  piece.  Note  the 
time  and  reading  of  the  circle,  then  quickly  bring  the  same 
wire  tangent  to  the  other  limb  of  the  sun.  Note  the  time 
and  reading  of  the  circle.  The  mean  of  the  times  and  read- 
ings will  be  those  corresponding  to  an  observation  of  the 
center.  Then  turn  the  instrument  in  azimuth  and  read  the 
circle  when  the  line  of  sight  is  on  the  terrestrial  object. 

In  case  only  one  limb  is  observed,  a  correction  must  be  ap- 
plied to  the  reading  of  the  circle  to  reduce  the  bearing  to  that 
of  the  center. 

In  Fig.  Ill,  this  correction  is  the  angle  SZS'  where  88' 
=.  x,  the  sun's  semi-diameter. 


ASTRONOMICAL  BEARING  457 

The  triangle  SZS'  being  right  angled  at  S'  and  90°  —  ZS 
being  equal  to  li,  the  true  altitude, 

sin  SS'      sin  S8'      sin  x 


sin  tjtjto   —  — ; — »~T?  — -    7~  — •  — 

sin  ZS        cos  h        cos 

Since  the  correction  and  semi-diameter  are  small, 

corr.  =  x  sec  hf  (175) 

the  sign  of  the  correction  depending  on  the  limb  observed. 

225.  Second  method. — In  this  case  the  true  azimuth  of  the 
heavenly  body  is  found  as  before,  the  astronomical  triangle 
being  solved  for  Z  by  either  the  time-azimuth  or  the  altitude- 
azimuth  method. 

Let  Fig.  110  represent  the  bodies  projected  on  the  plane 
of  the  horizon  and  the  triangles  involved ;  the  heavenly  body's 
true  place  is  Sf  its  apparent  place  M.  The  apparent  place  of 
the  terrestrial  object  is  0,  and  MO  is  the  observed  angular 
distance  of  the  object  0  from  the  heavenly  body's  center 
(that  is  the  sextant  reading  corrected  for  instrumental  errors, 
and  in  the  case  of  the  sun  for  semi-diameter). 

PZ8  is  the  astronomical  triangle,  PZ  the  co-latitude,  PS 
polar  distance,  ZS  co-altitude,  and  PZ8  the  azimuth. 

If  Z  is  not  gotten  from  a  time  azimuth,  it  is  gotten  from 
an  altitude  azimuth,  the  altitude  being  the  true  altitude  of 
M  found  by  observation  when  arc  M 0  is  measured.  Measure 
with  a  sextant  the  angular  distance  MO  between  the  bodies. 
At  the  time  of  measuring  the  arc  M 0,  note  the  time  or 
measure  the  altitude  of  M;  also  measure  the  altitude  of  0. 
Correct  the  altitudes  of  both  M  and  0  for  instrumental  errors 
and  dip,  thus  getting  the  apparent  altitudes  of  M  and  0. 
The  correction  for  dip  is  taken  from  Table  14,  Bowditch,  in 
case  of  a  free  horizon;  from  Table  15,  in  case  of  an  obstructed 
horizon. '  In  the  latter  case  this  correction  may  be  computed 
by  formula  (150),  Art.  208. 


458  NAUTICAL  ASTRONOMY 

When  the  true  altitude  is  found  from  a  time  azimuth,  it  is 
reduced  to  an  apparent  altitude  by  adding  the  refraction  and 
subtracting  the  parallax. 

Letting  h'  be  the  apparent  altitude  of  M, 
Q  '   "  "  0. 

there  are  given  in  the  triangle  MZO  the  three  sides: 
ZM=  .90°  —h', 
ZO  =  90°  --  Q, 
MO  =  D,  the  corrected  distance. 

To  find  MZO  =  £,  the  difference  of  azimuths. 
From  trigonometry, 


sin  »  MZO  = 


cos  Q  cos  h 


D  +  'Q  +  h' 
and  letting  --  -  -  =  s,  we  have 


cos     cos 
also  from  trigonometry, 


cos  /t  cos  Q 
and  letting  -  -  -  —  =  s,  we  have    : 


cos  s  cos  (5  —  D) 

(179) 


eo.vco.0 

*-  (180) 


COS 


Formula  (177)  is  preferable  when  £  is  <  90°,  (179)  when 
£  greatly  exceeds  90°. 

When  the  body  0  is  in  the  true  horizon,  Q  =  0°,  that  is  the 
observed  altitude  is  equal  to  the  dip;  in  Fig.  110,  aM  is  the 


ASTRONOMICAL  BEARING  459 

corrected  distance,  the  triangle  &Mm  is  right  angled  at  m  and 
by  Napier's  rules, 

cos  D  =  cos  li'  cos  aw, 
cos  am  =  cos  £  =  cos  D  sec  lir .  (181) 

or  in  (180),  if  Q  =  0, 

tan  J£=  V  [tan  4  (D  +  V)  tan  J  (D  —  V)]      (182) 

If  the  observed  object  0  is  exactly  in  the  water  line,  the  ap- 
parent altitude  is  equal  to  the  dip  and  is  negative,  or  Q 
=  (-)  dip. 

The  most  favorable  conditions  for  observation  are  when 
the  heavenly  body  is  on  the  prime  vertical  at  a  low  altitude 
and  the  distance  MO  approximates  90°;  the  ideal  condition 
being  when  both  bodies  are  in  the  true  horizon,  or  £  =  D. 

When  the  terrestrial  object  presents  a  vertical  line  to  which 
the  sun  may  be  brought  tangent,  the  sun's  diameter  through 
the  point  of  contact  0  will  not  be  in  the  direction  of  the  dis- 
tance OMf  but  perpendicular  to  the  vertical  circle  through  the 
terrestrial  object,  ZO,  and  a  correction  must  be  applied  to  the 
measured  distance  to  obtain  D.  It  is  obtained  from  the 
formula 

corr.  =  8  sin  M OZ  where  8  =  sun's  semi-diameter. 

The  altitude  of  0  is  very  small  anyhow,  and  by  consid- 
ering its  altitude  as  zero,  M  OZ  equals  MsiZ,  so  that  M  OZ  is 
found  with  sufficient  accuracy  from  the  right  triangle  Mma, 
taking  Ma  equal  to  the  uncorrected  distance,  that  is,  the  sex- 
tant arc  corrected  only  for  I.  C.  Letting  this  uncorrected 
distance  be  D',  we  have  by  Napier's  rules, 

sin  Ji'  —  sin  D'  sin  M  sun  =  sin  7X  cos  MaZ  9 
or,  sin  7i'  =  sin  D'  cos  MOZf 
cos  M  OZ  =  sin  h'  cosec  D'.  (183) 


460                           NAUTICAL  ASTRONOMY 

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HOUR  ANGLE  AND  LONGITUDE 


461 


Hour  Angle,  Local  Time,  and  Longitude. 

226.  To  find  the  hour  angle  of  a  heavenly  body  at  a  given 
place,  and  thence  the  local  time,  the  altitude  of  the  body  and 
the  Greenwich  time  being  known. 

Noting  the  time  of  observ- 
ing the  body's  altitude  by  a 
watch  compared  with  a  chro- 
nometer regulated  to  Green- 
wich mean  time,  the  G-.  M.  T. 
of  observation  is  found,  and 
for  this  the  declination  of  the 
body  is  taken  from  the  Nauti- 
cal Almanac.  Knowing  the  lati- 
tude and  reducing  the  observed  FIQS  n2 
to  a  true  altitude,  the  three 

sides  of  the  astronomical  triangle  are  known.     By  spherical 
trigonometry, 


sin    A 


•J! 


sin  (8  —  b)  sin  (8  —  c) 


sin  b  sin  c 
in  which  a,  b,  and  c  represent  the  three  sides  of  the  triangle 


Applying  this  formula  to  the  astronomical  triangle  PZM 
(Fig.  112),  and 
letting  A  =  t,  the  hour  angle  of  the  heavenly  body  ; 

a  =  90°  —  hf  the  complement  of  its  true  altitude; 

b  =  90°  —  d  =  p,  its  polar  distance; 

c  •=.  90°  —  L,  the  complement  of  the  latitude  ; 
then 


s  _  a  +  &  +  c  =  90°  —  h  -f  p  +  90°  —  L  __  9()0  __ 


8-  c=  90°  -  -    -  (90°  -L)  = 


462  NAUTICAL  ASTRONOMY 

Therefore, 


=     lr 

if  L 


pBnp  —      -i 
cos  £  sin  p 
Now  letting  5  =  $(L  +  h  +  p),  then  %(L  +  p  —  h)  =  s—h, 


and  sin  £  t  =     I  ["cosssm  (s~  h)  "]  (184) 

\  L       cos  L  sin  ;?      J 

In  like  manner  may  be  deduced  from  an  application  to  the 

sin  8  sin  (8  —  a) 

triangle  of  the  formula  cos2  $  A  = : — = — : •- 

sm  b  sm  c 

the  following: 

cos  £  t  =     I  rsm(g~~     )cos(g  —  ffjn          ^  8g  ^ 

\  L  cos  L  sm  p 

where  5  =  J  (L  +  ft  +  ^)- 

The  above  formulae  are  adapted  to  use  with  any  tables,  but 
for  those  navigators  supplied  with  "  Inman's  Tables  "  or  any 
tables  of  log  haversines  and  one-half  log  haversines,  the  fol- 
lowing deduction  and  formula  will  be  found  of  interest.  It 
is  largely  used  in  the  British  Navy,  and  has  many  advantages. 
From  trigonometry, 


J  A  -  a  =  z 

sm  &  sm  c  I  5  =±  90    —  rf, 


L-\-d  —  z 
2 


—  J  =  90°  —  -'        ~:i:  _  (90°  4-  rf)  = 


-  e  =  90°  -  L±°^J  -  (90°  -  £)  =  L  +  *  ~ 
2 


8n  . 


Therefore,  sin2  -|  ^  ^  -  -T—  j-y 

cos  a  cos  L 


HOUR  ANGLE  AND  LONGITUDE  463 

Now 
n  2  ±t  =  haversine  t,  sin  —  :^4^   -  =v'haver  (  z  —  (L—  d)), 


and  sin  —       =  ^  haver  («+  (L  —  flf)). 

A 

Therefore,  using  logs  : 

log  haver  tf  =.  J  log  haver  (2  —  (L  —  d}  ) 

+  J  log  haver  (2  +  (L  —  d)  )        (186) 
+  log  sec  d  +  log  sec 

When  t  greatly  exceeds  6  hours,  as  is  often  the  case  in  high 
latitudes,  it  should  be  found  from  formula  (185). 
When  L  =  90°,  the  zenith  is  at  the  pole; 

in  (184)  p  +  h  =  90°  and 


cos  L 

Therefore,  in  very  high  latitudes  it  is  impracticable  to  find 
with  exactness  the  local  time  as  the  formulae  for  hour  angles 
then  approach  the  indeterminate  form. 

The  formulae  also  reduce  to  the  indeterminate  form  when 
d  =  90°,  at  which  time  the  star  would  be  at  the  pole  and, 
therefore,  its  altitude  would  be  the  latitude  ;  for  this  reason 
when  working  for  time  avoid  stars  of  very  large  declination. 

For  time,  bodies  should  be  observed  when  on  or  near  the 
prime  vertical  (Art.  237),  and  the  desirability  of  this  position 
increases  as  the  latitude  increases.  In  latitudes  beyond 
66°  30'  an  error  of  1'  in  the  altitude  will  cause  an  error  of  at 
least  10s  of  time  in  the  longitude. 

Using  formula  (184),  it  is  not  necessary  to  take  out  •§  t  in 
arc,  then  multiply  it  by  2,  and  convert  it  into  time.  In  Table 
44,  Bowditch,  t  may  be  taken  directly  from  the  p.  m.  column 
corresponding  to  log  sin  J  t  in  the  sine  column, 

In  taking  out  hour  angles,  take  them  from  the  p.  m.  column, 


464  NAUTICAL  ASTRONOMY 

marking  them  +  when  the  body  is  West  of  the  meridian,  ( — ) 
when  the  body  is  East  of  the  meridian. 

When  the  body  is  the  sun,  t  its  hour  angle  is  local  appar- 
ent time  when  the  sun  is  West  of  the  meridian;  but  if  the 
sun  is  East  of  the  meridian,  its  H.  A.  is  ( — )  i,  and  the  local 
apparent  time  is  24  hours  —  i.  This  is  astronomical  L.  A.  T. 
Thus,  if  the  sun's  H.  A.  is  +  4  hours,  the  L.  A.  T.  is  4  p.  m. ; 
if  the  sun's  H.  A.  is  ( — )  4  the  L.  A.  T.  is  20  hours  astro- 
nomically, or  8  a.  m.  civil  time,  that  is  to  say,  if  the  sun's 
H.  A.  is  ( — )  4  hours,  the  sun  will  not  be  on  the  meridian 
for  4  hours. 

Having  the  H.  A.  of  the  sun  which  is  L.  A.  T.  or  24  hours 
-  L.  A.  T.,  to  obtain  local  mean  time,  the  equation  of  time 
must  be  taken  out  of  the  Nautical  Almanac  for  the  Greenwich 
instant  and  applied  with  its  proper  sign. 

Then  the  difference  between  this  local  mean  time  and  the 
corresponding  Greenwich  mean  time  will  be  the  longitude; 
West  if  the  Greenwich  time  is  the  greater,  otherwise  East. 

Conditions  of  observation. — A  little  further  on  (Art.  237), 
it  will  be  shown  that  altitudes  for  time,  whether  of  the  sun, 
moon,  a  planet,  or  a  star  should  be  taken  when  the  body  is  on 
or  near  the  prime  vertical,  and  certainly  more  than  45°  and 
less  than  135°  in  azimuth.  The  altitude  should  be  sufficiently 
high  to  eliminate  errors  of  refraction,  say  above  10°,  and 
especially  so  when  refraction  may  be  affected  by  fog  or  mist. 

It  should  be  a  rule,  when  observing  heavenly  bodies,  to  take 
several  altitudes  in  quick  succession;  and  the  mean  of  3  or  5 
altitudes,  thus  taken  and  so  selected  that  the  differences  of 
altitude  vary  with  the  differences  of  time,  should  be  used  in 
preference  to  a  single  observation.  Whenever  the  sun  is  ob- 
served for  time,  its  compass  bearing  should  be  observed  for 
compass  error  and  the  heading  of  the  ship  per  compass  also 
carefully  noted. 


TIME  SIGHT  OF  SUN  465 

Such  sights  worked  for  time  and  longitude  are  known  as 
"  time  sights." 

227.  Rules  for  working  a  time  sight  of  the  sun. — (1)  Find 
the  Greenwich  mean  time  and  date.  It  is  shown  in  the  col- 
umn marked  "  Times,"  in  the-  form  for  work  following,  how 
the  G.  M.  T.  of  observation  is  obtained.  Applying  the  chro- 
nometer comparison  (C — W)  to  the  watch  time  of  observa- 
tion gives  the  chronometer  time  of  observation,  and  if  to  this 
is  applied  the  chronometer  correction  on  G.  M.  T.  when  leav- 
ing port  brought  up  to  date  for  daily  loss  or  gain,  the  result 
will  be  the  G.  M.  T.  of  observation,  but  care  must  be  taken  to 
see  that  this  time  is  astronomical  time,  and  that  the  date  is 
correct. 

(2)  Reduce  the  sextant  altitude  to  the  true  altitude  of  the 
center,  and  take  from  the  Nautical  Almanac  for  the  Green- 
wich mean  noon  of  the  given  astronomical  day  the  sun's  de- 
clination, H.  D.  of  declination,  equation  of  time,  H.  D.  of 
the  equation  of  time,  and  correct  both  declination  and  equa- 
tion of  time  for  the  G.  M.  T.     If  the  declination  is  of  the 
same  name  as  the  latitude,  find  p  =  90°  —  d;  if  the  declina- 
tion is  of  a  different  name  from  the  latitude,  find  p  =.  90° 
+  d.     Note  whether  the  equation  of  time  is  +  or  ( — )  to 
apparent  time. 

(3)  Combine  h,  L  and  p  as  required  in  the  equation  (184) 
and  as  per  form  illustrated  in  example  on  page  467.    Having 
found  the  log  sin  -J  t,  look  for  it  in  the  column  of  sines  (Table 
44,  Bowditch),  and  take  out  the  corresponding  time  from  the 
a.  m.  or  p.  m.  column  according  as  the  sight  is  an  a.  m.  or 
p.  m.  sight.     This  quantity  is  the  local  civil  apparent  time. 
The  time  from  the  p.  m.  column  is  also  astronomical  time, 
but  12  hours  must  be  added  to  the  reading  from  the  a.  m. 
column  to  reduce  it  to  astronomical  time.     Applying  the  equa- 
tion of  time,  with  the  proper  sign,  to  the  local  apparent  time 
gives  the  local  mean  time  of  observation. 


i66  NAUTICAL  ASTRONOMY 

(4)  The  difference  between  the  local  and  Greenwich  mean 
times  of  observation  is  the  longitude  in  time;  West  if  the  local 
time  is  less  than  the  Greenwich  time,  otherwise  East  (Art. 
179). 

(5)  Or,  the  G.  A.  T.  may  be  found  by  applying  the  equa- 
tion of  time,  with  the  proper  sign,  to  the  G.  M.  T.;  then  the 
longitude  will  be  the  difference  between  the  G.  A.  T.  and 
L.  A.  T.,  West  if  G.  A.  T.  is  the  greater,  East  if  the  L.  A.  T. 
is  the  greater. 

228.  Time  sights  of  the  moon,  a  planet,  or  a  star. — When 
the  hour  angle  determined  by  the  formulae  of  Art.  226  is  that 
of  any  other  heavenly  body  than  the  sun,  that  is,  of  the  moon, 
a  planet,  or  a  star,  the  right  ascension  of  the  body  must  be 
taken  out  of  the  Nautical  Almanac  for  the  Greenwich  in- 
stant (in  the  case  of  stars,  for  the  Washington  instant) ;  then 
the  algebraic  sum  of  the  hour  angle  and  right  ascension  of 
this  body  will  give  the  local  sidereal  time.  Having  con- 
verted the  G.  M.  T.  into  the  corresponding  G.  S.  T.  (Art. 
192),  the  difference  between  the  G.  S.  T.  and  L.  S.  T.  will  be 
the  longitude,  West  if  the  G.  S.  T.  is  the  greater;  East,  if  the 
L.  S.  T.  is  the  greater  (Art,  179). 


A.  M.  TIME  SIGHT  OF  SUN 


467 


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TIME  SIGHT  OF  A  STAR  469 

229.  Rules  for  working  a  time  sight  of  a  star. — (1)  When 
observed  for  time,  a  star  should  be  on  or  near  the  prime  verti- 
cal to  give  the  best  results.  The  observation  should  be  made 
when  the  horizon  is  well  defined  during  twilight,  or  when  the 
moon  is  shining. 

(2)  Find  the  correct  G.  M.  T.  and  date. 

(3)  For  this  G.  M.  T.  take  from  the  Nautical  Almanac  the 
R.  A.  M.  O  (Art.  185),  and  convert  the  G.  M.  T.  into  G.  S.  T. 

(4)  Look  for  the  star's  approximate  right  ascension  in  the 
mean  place  table  of  fixed  stars;  then  in  the  apparent  place 
table  find  the  star's  R.  A.  and  declination  to  the  nearest  sec- 
ond of  arc.     For  use  in  a  time  sight  they  require  no  correction. 

(5)  Reduce  the  sextant  altitude  to  a  true  altitude.     Sub- 
stitute h,  L,  and  p  in  the  formula,  as  per  form  following,  and 
having  found  the  log  sin  \  t,  look  for  it  in  the  log  sine  col- 
umn of  Table  44,  Bowditch,  and  abreast  of  it  in  the  column 
marked  "  Hour  p.  m."  will  be  found  the  star's  H.  A.  or  t  in 
hours,  minutes,  and  seconds  of  time.     The  exact  value  of  t 
may  be  found  by  interpolation  or  by  using  the  table  of  pro- 
portional parts  at  the  foot  of  the  page. 

(6)  If  the  star  is  East  of  the  meridian,  mark  t  (—);  if 
the  star  is  West  of  the  meridian,  mark  t  +.     To  the  hour 
angle  t  apply  the  star's  R.  A.;  the  algebraic  sum  will  be  the 
L.  S.  T.,  the  difference  between  which  and  the  G.  S.  T.  will 
be  the  longitude,  West  if  the  G.  S.T.is>  L.  S.  T.,  or  East  if 
G.  S.T.is<  L.  8.  T. 


470 


NAUTICAL  ASTRONOMY 


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a.  2 

TIME  SIGHT  or  THE  MOON  471 

230.  Rules  for  working  a  time  sight  of  the  moon. — (1) 

Proceed  exactly  as  with  a  time  sight  of  a  star  with  differences 
noted  below.  The  moon's  right  ascension  and  declination  are 
tabulated  for  each  day  and  hour  of  G.  M.  T.  with  differences 
for  one  minute,  and  are  taken  out  and  corrected  accordingly. 
The  moon's  S.  D.  and  H.  'P.  are  taken  from  page  IV  of  the 
Nautical  Almanac.  The  augmentation  of  the  moon's  8.  D. 
is  taken  from  Table  IS,  Bowditch,  and  is  +  or  ( — ),  depend- 
ing on  whether  the  observed  limb  was  the  lower  or  upper  limb 
of  the  moon.  In  refined  observations  a  reduction  from  Table 
19,  Bowditch,  is  applied  to  the  H.  P.  Since  the  radius  from 
the  center  of  the  earth  to  an  observer  in  high  latitudes  is  less 
than  the  equatorial  radius,  and  the  tabulated  H.  P.  is  for  an 
observer  on  the  equator,  the  H.  P.  should  be  reduced  for  the 
latitude,  the  reduction  being  siibtr  active.  In  work  at  sea  this 
reduction  is  usually  omitted. 

(2)  Reduce  the  sextant  altitude  of  the  moon's  limb  to  the 
apparent  altitude  of  the  moon's  center  by  applying  the  first 
correction,  which  consists  of  (±  augmented  S.  D.  ±  I.  C.  — 
Dip}.     Then  with  this  apparent  altitude  of  the  moon's  cen- 
ter and  the  H.  P.  find  from  Table  24,  Bow  ditch,  the  second 
correction  consisting  of  parallax  and  refraction.     Add  this 
to  the  apparent  altitude  and  the  result  will  be  the  true  alti- 
tude of  the  moon's  center  (see  Ex.  152,  Art.  212). 

(3)  Then  proceed  as  in  a  time  sight  of  a  star. 


472 


NAUTICAL  ASTRONOMY 


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TIME  SIGHT  OF  A  PLANET  473 

The  reliability  of  a  moon  sight  for  time.— The  value  of 
any  observation  for  longitude  depends  upon  the  correctness  of 
the  altitude,  the  latitude,  and  the  declination.  An  error  in 
the  G.  M.  T.  produces  an  equal  error  in  the  longitude  besides 
causing  an  error  in  the  declination  of  any  body  whose  decli- 
nation changes  rapidly.  In  the  case  of  the  moon,  not  only 
the  declination  but  the  right  ascension  (an  error  in  which 
will  affect  the  longitude)  changes  rapidly,  so  the  G.  M.  T. 
should  be  accurately  known  to  ensure  good  results.  For  these 
reasons  a  time  sight  of  the  moon  is  not  so  desirable  as  one  of 
the  sun  or  of  a  star. 

231.  Rules  for  working  a  time  sight  of  a  planet. — (1)  Pro- 
ceed exactly  as  with  a  time  sight  of  a  star,  remembering, 
however,  to  correct  the  right  ascension  and  declination  of  a 
planet  for  G.  M.  T. 

(2)  The  planets  have  a  semi-diameter  and  parallax  which, 
for  theoretical  reasons,  will  be  used  in  the  corrections  to  the 
altitude  in  the  example  that  follows,  though  it  would  be  a 
useless  refinement  to  use  them  in  sea  observations.    At  sea,  a 
planet's  altitude  should  be  corrected  for  I.  C.,  dip,  and  re- 
fraction. 

(3)  Proceed  otherwise  as  if  with  a  time  sight  of  a  star. 

232.  Haversine  formula. — The  hour  angle  t  may  be  found 
from  (157)  or  that  formula  transformed  into  one  containing 
haversines  only,  but  (186)  will  permit  of  a  more  convenient 
method  of  solution  and  a  better  arrangement  of  form  than 
either  of  these.    See  Exs.  170(a)  and  170(b). 

For  p.  m.  solar  sights  the  astronomical  L.  A.  T.  (or  t)  is 
taken  from  the  top  of  the  page  of  haversine  tables ;  for  a.  m. 
solar  sights,  the  L.  A.  T.  (or  24h  —  t)  is  taken  directly  from 
the  bottom  of  the  page.  In  case  of  the  moon,  a  star,  or  a 
planet,  the  hour  angle  t  is  taken  from  the  top  of  page  and 
marked  -(-  or  --  according  as  the  body  is  W.  or  E.  of  the 
meridian. 


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TIME  OF  SUNSET  475 

233.  Sunrise  or  sunset  time  sights. — When  the  sun's  cen- 
ter is  in  the  visible  horizon,  East  or  West,  its  true  altitude  or 
h  equals  —   (refraction   +   dip  —  parallax).     The  watch 
time  of  this  instant  may  be  found  by  noting  the  watch  times 
when  the  lower  and  upper  limbs  are  in  the  visible  horizon, 
just  appearing  at  sunrise  or  disappearing  at  sunset,  and  tak- 
ing their  mean. 

Having  the  watch  time  of  the  instant,  the  C — W,  and  the 
chronometer  error  on  G.  M.  T.,  proceed  to  work  a  time  sight 
of  the  sun  with  a  negative  altitude  which  numerically  equals 
(refraction  +  dip  —  parallax),  the  refraction  and  parallax 
being  for  an  altitude  of  0°  and  the  dip  depending  on  the 
height  of  the  eye. 

However,  owing  to  the  difficulty  of  noting  the  times  at 
contact  of  limbs  with  the  horizon  and  the  uncertainties  of 
refraction,  the  result  should  be  regarded  as  only  approximate; 
and  this  method  should  be  used  only  when  fog  or  cloudy 
weather  has  prevented,  or  may  prevent,  the  navigator  from 
getting  more  reliable  observations  of  the  sun  or  stars. 

234.  Time  of  sunset. — The  instant  of  sunset  is  when  the 
sun's  upper  limb  is  just  disappearing  below  the  visible  hori- 
zon, or  when  h  ==  —  (refraction  +  dip  +  S.  D.  —  paral- 
lax).    For  this  altitude,  a  given  latitude,  and  declination,  the 
hour  angle  of  the  sun  may  be  found  by  the  time-sight  formula. 
This  hour  angle  will  be  the  civil  local  apparent  time  of  sun- 
set; (12  hours  —  the  H.  A.)  will  be  the  L.  A.  T.  of  sunrise. 

The  L.  M.  T.  of  sunset  may  be  found  by  applying  the  equa- 
tion of  time  for  the  instant  to  the  L.  A.  T.  of  sunset. 

As  the  declination  and  equation  of  time  are  tabulated  for 
Greenwich  time,  to  find  the  declination  at  the  instant  of  sun- 
set, it  will  be  necessary  to  either  assume,  or  take  from  azimuth 
or  sunset  tables,  an  approximate  time  of  sunset,  apply  the 
longitude,  and  obtain  an  approximate  Greenwich  time  of  sun- 
set for  which  the  declination  mav  be  found. 


NAUTICAL  ASTRONOMY 


For  the  equation  of  time  it  is  better  to  proceed  thus :  Having 
found  by  computation  in  the  time  sight  the  L.  A.  T.  of  sun- 
set, find  the  correct  G-.  A.  T.  by  applying  the  longitude,  and 
for  this  G-.  A.  T.  take  from  the  Nautical  Almanac  the  equa- 
tion of  time  which,  if  applied  to  the  L.  A.  T.  of  sunset,  will 
give  the  required  L.  M.  T. 

The  L.  M.  T.  of  sunrise  and  sunset  may  be  found  in  Table 
10,  Bowditch,  or  in  the  Tide  Tables  issued  by  the  U.  S.  C. 
and  G-.  Survey. 

235.  To  find  the  duration  of  twilight.-— Twilight  lasts  till 
the  sun  has  sunken  18°  below  the  visible  horizon,  and  there 

will  be  continual  light  so  long  as 
the  sun  at  its  lower  transit  is  not 
more  than  18°  below  the  visible 
horizon  at  the  place. 

The  difference  of  hour  angles 
of  the  sun  obtained  by  time  sights, 
using  altitudes  of  ( — )  (ref.  -f- 
dip  +  S.  D.  --  p)  and  of  (— ) 
(18°  +  ref.  +'  dip  +  S.  D.  —  p) 
will  give  the  duration  of  twilight. 

236.  To  find  the  hour  angle  of  a  heavenly  body  when  in 
the  horizon. — In  Fig.  113,  H  is  the  body  in  the  horizon,  its 
H.  A.  is  HPZ  —  t  and  L  HPN  —  180°  --  i,  PN  —  L, 
PH  =  90°  —  d,  and  L  PNH  =  90°. 

By  Napier's  rules,  cos  HPN  =  tan  PN  cot  PH, 

or  cos  t  —  —  tan  L  tan  J,  (187) 

and  if  Hs  <  6  hrs.,  2t  will  be  <  12  hrs. ;  also  if  J  is  >  6  hrs., 
2t  will  be  >  12  hrs. 

From  the  above  it  is  apparent  that  bodies  of  positive  decli- 
nation (same  name  as  latitude)  will  be  above  the  true  hori- 
zon for  more  than  12  hours,  bodies  of  negative  declination 
will  be  above  the  horizon  less  than  12  hours.  This  interval 


FIG.  113. 


EFFECT  OF  ERRORS  IN  DATA  477 

2t  is  for  the  sun  an  interval  of  apparent  time;  for  a  fixed 
star,  an  interval  of  sidereal  time. 

Length  of  day  and  night. — As  the  length  of  the  day  is  de- 
termined by  the  length  of  time  the  sun  is  above  the  horizon  of 
a  place,  it  is  evident  that  since  cos  t  =  —  tan  d  tan  L,  at  the 
equator  where  L  =  0,  cos  t  =  0  and  t  =  6  hours,  so  that  at 
the  equator  the  day  .is  12  hours  long  in  all  seasons.  At  the 
equinoxes  d  =  0,  tan  d  =  0,  and  2t  =  12  hours,  so  that  when 
the  sun  is  at  the  equinoxes^  the  day  is  everywhere  12  hours. 

Within  the  Arctic  Circle,  if  d  —  +  23J°  and  L  =  66J°, 
cos  t  =  —  1,  2t  =  24  hours,  and  there  is  no  night;  this 
would  be  the  case  of  midsummer  in  latitudes  beyond  66J°. 
If  d  =  —  23£°  and  L  =  +  66J°,  cos  t  will  be  +  1,  2t  will 
be  0  hours,  and  there  will  be  no  day,  only  night,  as  in  the  case 
of  midwinter  in  latitudes  beyond  66  J°.  Thus  it  is  plain  that 
12  hours  is  the  average  length  of  a  day  throughout  a  year; 
on  a  given  date  when  the  sun's  declination  is  positive,  the  day 
is  >•  12  hours,  and  on  a  day  six  months  later,  when  the  sun's 
declination  is  numerically  the  same  but  negative,  the  day  will 
fall  equally  short  of  12  hours. 

237.  Effect  of  small  errors  in  data. — In  the  solution  of 
the  astronomical  triangle  for  time,  and  hence  for  longi- 
tude, the  elements  involved  are  the  declination  from  the 
Nautical  Almanac  which  may  be  regarded  as  accurately 
known,  a  measured  altitude  which  is  affected  by  errors 
of  observation,  errors  of  the  instrument,  and  errors  of 
refraction,  and  a  latitude  by  observation  or  dead  reckoning 
which  depends  not  only  on  accuracy  of  original  determination 
by  observation  but  on  the  correctness  of  course  and  distance 
run,  etc.  Thus  it  is  apparent  that  both  the  altitude  and 
latitude  are  liable  to  error,  and  it  is  desirable  to  consider  the 
effect  on  the  resulting  longitude  of  (1)  a  small  error  in  alti- 
tude, (2)  a  small  error  in  latitude;  (3)  to  find  the  position  of 
a  body  when  its  altitude  changes  most  rapidly,  and  then  to 


•478  NAUTICAL  ASTRONOMY 

determine  the  most  favorable  position  of  a  heavenly  body  for 
observations  for  time  or  longitude. 

By  substitution  in  the  trigonometric  formula 
cos  a  =  cos  1}  cos  c  +  sin  b  sin  c  cos  A, 

letting  t  =  A,  90°  —h  =  afW°  —  d  =  I,  and  90°  —L  =  c. 
we  have  the  fundamental  equation 

sin  li  r=  sin  L  sin  d  +  cos  L  cos  d  cos  £. 
By  differentiation,  h  and  tf  variables, 

cos  hdh  =  —  cos  L  cos  d  sin  tdt, 

dt___ cos  ft 

dh          cos  L  cos  d  sin  £ ' 

but  sin  t  =  sin  Z  cos  A  sec  d;  therefore, 

*  =  -  -   -= -4^| =—  -.  =  - sec  L  cosec  Z.  (188) 

^A  cos  L  cos  d  sin  ^  cos  h  sec  d 

Differentiating  the  fundamental  equation,  L  and  t  variables, 

dt       sin  d  cos  L  —  cos  d  cos  t  sin  7v 

we  have  -ry  = = — : — 

dL  cos  d  cos  L  sin  tf 

From  trigonometry, 

sin  a  cos  B  =  cos  &  sin  c  —  sin  &  cos  c  cos  A. 

If  *  =  A,  B  =  Z,  a  =  90°  —  h,  I  =  90°  '—d,c  =  90°  —  L, 
by  substitution,  we  have 

cos  h  cos  Z  =  sin  d!  cos  L  —  cos  J  sin  L  cos  £, 
and,  by  rule  of  sines,  sin  t  =  sin  Z  cos  h  sec  d. 
Making  these  substitutions  in  equation  for  -=-=     we  have 

*  J^Af^-  ,=  cot  Z  seel          (189) 
dL     cos  d  cos  L  sin  Z  cos  /i  sec  d 

(188)  shows  H  to  be  least  when  L  —  0°  and  Z  =  90°. 

(189)  shows  J^,to  be  least  when  L  =  0°  and  Z  =  90°. 


EFFECT  OF  ERRORS  IN  DATA  479 

To  find  the  position  of  a  body  in  azimuth  when  its  altitude 
changes  most  rapidly. 
In  Art.  222,  by  differentiation,  we  found  formula  (167) 

dZ  _  si11  L  cos  ^  —  cos  L  sin  h  cos  Z 

dh  cos  L  cos  h  sin  Z 

4£  =tan  L  cosec  Z  —  tan  h  cot  Z. 
an 

Therefore,  dh  =  z = J^—r  (19°) 

tan  L  cosec  Z  —  tan  h  cot  ^ 

(190)  shows  dh,  or  the  change  in  altitude,  to  be  greatest 
when  Z  •=.  90°;  or,  in  other  words,  when  the  heavenly  body 
is  on  the  prime  vertical. 

Conclusions. —  ( 1 )  Considering  the  effects  of  errors  in  h  and 
L,  sights  for  longitude  are  best  in  low  latitudes. 

(2)  An  error  in  altitude,  or  an  error  in  latitude,  will  pro- 
duce the  least  change  in  the  hour  angle  when  the  heavenly 
body  is  on  the  prime  vertical. 

(3)  The  motion  of  a  heavenly  body  in  altitude  is  most 
rapid  when  it  is  on  the  prime  vertical;  its  altitude  can  be 
taken  with  greater  accuracy;  and  when  the  diurnal  circle  of 
a  body  corresponds  to  the  prime  vertical,  the  change  of  alti- 
tude is  directly  proportional  to  the  change  in  time,  and  the 
mean  of  a  number  of  altitudes  will  correspond  to  the  mean 
of  their  times  of  observation. 

For  these  reasons  it  is  better  to  observe  a  heavenly  body 
when  on  or  near  the  prime  vertical  when  observing  for  time 
or  longitude. 

When  the  latitude  and  declination  are  of  the  same  name, 
there  is  no  difficulty  in  observing  the  body  near  the  prime 
vertical  and  at  an  altitude  sufficiently  great  to  eliminate  the 
uncertainties  of  refraction.  In  low  latitudes,  when  L  and  d 
are  of  the  same  name,  the  body  may  be  on  the  prime  vertical 
when  only  a  few  minutes  from  the  meridian,  in  the  case  of 
the  sun  near  noon,  and  still  be  available  for  time  observations. 


480  NAUTICAL  ASTRONOMY 

These  remarks  should  emphasize  the  fact  that  the  suita- 
bility of  heavenly  bodies  for  time  observations  depends  more 
on  the  azimuths  than  on  the  hour  angles  of  such  bodies. 

When  L  and  d  are  of  different  names,  the  diurnal  circles 
do  not  cross  the  prime  vertical  above  the  horizon,  and,  under 
such  circumstances,  bodies  are  nearest  to  the  prime  vertical 
when  in  the  horizon;  and  such  bodies  should  be  observed,  if 
necessary  to  observe  them  for  time,  as  soon  as  the  altitude  is 
sufficiently  high  to  be  unaffected  by  errors  of  refraction — that 
is  at  least  10°. 

By  the  time  the  sun  has  reached  a  proper  altitude  for  ob- 
servations in  winter  time,  it  is  so  far  from  the  prime  vertical 
that  any  error  in  altitude  or  latitude  will  produce  a  larger 
one  in  longitude,  and  this  error  will  increase  with  the  latitude. 
However,  during  twilight  or  moonlight,  in  winter,  there  need 
be  no  difficulty  in  finding  suitable  stars  on  or  near  the  prime 
vertical  from  which  to  obtain  reliable  determinations  of  longi- 
tude. 

An  inspection  of  the  azimuth  tables  will  indicate  for  a 
given  latitude  and  declination  the  hour  angle  of  a  body  when 
on,  or  nearest  to,  the  prime  vertical,  and  from  it  the  local 
time  may  be  found. 

Since  when  on  the  prime  vertical,  a  heavenly  body  is  fall- 
ing or  rising  most  rapidly,  and  the  changes  of  altitude  are 
proportional  to  the  changes  of  time,  the  effect  of  an  error  of 
1'  in  the  altitude  on  the  resulting  hour  angle,  and  hence  longi- 
tude, can  be  gotten  by  dividing  the  difference  of  any  two  of 
the  recorded  times  of  observation  by /the  difference  of  the  cor- 
responding altitudes  in  minutes  of  arc.  The  result  will  be  in 
minutes  or  seconds  of  time  as  the  differences  of  times  are  in 
minutes  or  seconds  of  time. 

238.  The  practical  way  of  finding  the  effect  of  an  error  of 
1'  in  the  latitude,  on  the  resulting  longitude,  is  to  find  the 
longitudes  for  two  latitudes  differing,  say  10'  or  20',  then  di- 


WHEN  ON  OR  NEAREST  P.  V.  481 

vide  the  difference  of  longitude  by  the  difference  of  assumed 
latitudes  in  minutes  of  arc.  A  study  of  Sumner's  method 
will  make  this  plain. 

In  Table  I  of  this  book  will  be  found  tabulated  the  changes 
in  longitude  for  a  change  of  1'  of  latitude,  the  arguments 
being  the  observer's  latitude  and  the  body's  true  azimuth. 

239.  To  find  the  hour  angle  of  a  heavenly  body,  its  true 
altitude,  and  azimuth,  when  nearest  to  or  on  the  prime  ver- 
tical, that  is  nearest  in  azimuth  to  90°  (Figs.  A  and  B). 

There  are  seven  cases  that  may  be  considered: 

(1)  +  d  >  L.     It  is  evident  that  the  azimuth  will  be 
greatest  when  the  vertical  circle  is  tangent  to  the  diurnal 
circle  as  at  M  in  PZM  and  at  M±  in  PZM^;  M  =  90°, 
MI  =.  90°,  cos  t  =  tan  L  cot  d,  sin  h  =  sin  L  cosec  df 
sin  Z  =.  cos  d  sec  L.     The  body  is  circumpolar  when  d  equals 
or  is  >  co.  L  as  in  the  case  of  a  body  whose  diurnal  circle  is 
NM. 

(2)  +  d  =  L.     Here  the  diurnal  circle  is  tangent  to  the 
prime  vertical  at  the  zenith.     At  the  point  sought,  Z  =  90°. 
The  body  is  on  the  meridian  and  t  =  0.     The  body  is  in  the 
zenith  and  h  =  90°. 

(3)  +  d  <  L.     The  diurnal  circle  crosses  the  prime  ver- 
tical, and,  at  the  moment  of  crossing,  Z  =  90°  in  triangle 
PZMS ;  cos  t  =  cot  L  tan  df  sin  K  =  sin  d  cosec  L. 

(4)  d  =  0.     Here  the  diurnal  circle  is  in  the  plane  of  the 
equinoctial  and  passes  through  the  East  and  West  points  of  the 
horizon,  as  does  also  the  prime  vertical;  those  are  the  points 
sought.    Triangle  PZM4  ,  Z  —  90°,  cos  t  =  cot  L  tan  d  =  0, 
and  t  =  6  hours;  sin  h  =  sin  d  cosec  L  =  0,  and  h  =  0. 

(5)  —  d  <  L.     In  this  case,  d  is  of  a  different  name  from 
L.     The  diurnal  circle  intersects  the  prime  vertical  below  the 
horizon.     In  triangle  PZM5   (Fig.  B),  Z  —  90°,  cos  t  = 
cot  L  tan  ( —  d),  cos  t  is  negative,  and  t  is  >  6  hours;  sin  h 
—  cosec  L  sin  ( —  d},  h  is  negative,  and  the  body  is  below 
the  horizon. 


482 


NAUTICAL  ASTRONOMY 


FIG.  B, 


WIIKX  ON  OR  NEAREST  P.  V.  483 

The  nearest  point  to  the  prime  vertical  at  which  the  body 
is  visible  is  IP5  (Fig.  A),  when  the  body  is  in  the  horizon. 
From  the  triangle  PNM'5  (Fig.  A),  we  have  —  cos  t  = 
tan  L  tan  ( —  d),  or  cos  t  =  tan  L  tan  d  and  £  is  <  6  hours ; 
cos  NM'K  =  cos  Z  —  sec  L  sin  ( —  d),  cos  Z  is  negative  and 
Z  >  90°. 

(6)  —  d  =  L.     In  this  case,  the  diurnal  circle  is  tangent 
to  the  prime  vertical  at  the  nadir.     The  triangle  is  PZMn 
(Fig.  B),  Z  —  90°,  cos  t  —  cot  L  tan  (—d)  =  —  1,  cos  t 
—  —  1  and  t  =  12  hours,  sin  li  =  cosec  L  sin   ( —  d) 

-  1,  or  h  =  —  90°. 

The  nearest  point  to  the  prime  vertical  at  which  the  body 
is  visible  is  M'Q  (Fig.  A),  where  the  body  is  in  the  horizon. 
From,  the  triangle  PNM'Q ,  —  cos  t  —  tan  L  tan  ( —  d),  or 
cos  t  =  tan  L  tan  d,  t  is  <  6  hours;  cos  NM'Q  =  cos  Z 
•=..  sec  L  sin  ( —  d),  cos  Z  is  negative,  and  Z  is  >  90°. 

(7)  -  -  d  >  L.     Triangles  PZM7  and  PZMa   (Fig.  B), 
Z  >  90°  from  the  elevated  pole.     At  M7  and  M 8 ,  %  is  a 
minimum,  estimated  from  the  depressed  pole,  at  elongation. 
The     vertical     circle     is    tangent    to    the     diurnal     circle. 
M7  —  90°,  M8  =  90°.     Therefore,  in  triangles  PZM,  and 
PZMS ,  right  angled  at  M7  and  M 8 ,  cos  t  =  tan  L  cot  ( —  d), 
cos  t  is  negative,  and  t  is  >  6  hours;  sin  h  =  sin  L  cosec 
( —  d)  and  h  is  negative ;  sin  Z  =  sec  L  cos  d,  and  should  bo 
taken  in  the  second  quadrant. 

If  —  d  equals  or  is  >  co.  L,  the  body  is  never  visible  as 
in  triangle  PZM8  (Fig.  B) .  If  the  body  is  visible,  then  —  d  is 
<  co.  L  and  a  point  in  the  horizon  will  be  the  nearest  point 
to  the  prime  vertical  at  which  the  body  is  visible,  as  J/'7 , 
(Fig.  A).  In  the  triangle  *PNM'79  t  is  <  6  hours  and  Z  is 
>  90°,  as  in  cases  (5)  and  (G)  when  the  body  is  visible. 

Recapitulation  for  hour  angle  when  the  body  is  visible  and 
on  or  nearest  to  the  prime  vertical, 


484  NAUTICAL  ASTRONOMY 

When  d  is  of  the  same  name  as  L. 

If  d  is  <  L,  use  formula  cos  t  =  cot  L  tan  d.     (194) 

.  If  d  is  >  L,  use  formula  cos  t  —  tan  L  cot  d.     (195) 

'  When  d  and  L  are  of  different  names,  and  if  the  body  is  ever 

above  the  horizon,  it  will  be  nearest  to  the  prime  vertical  on 

rising  or  setting,  then  use  formula 

cos  t  =.  —  tan  L  tan  d.  (196) 

When  +  d  =  L,  the  diurnal  circle  passes  through  the 
zenith,  so  that  when  the  declination  is  equal  to,  or  nearly  equal 
to,  the  latitude,  observations  for  time  may  be  made  within  a 
few  minutes  of  meridian  passage,  the  mean  of  altitudes  cor- 
responding to  a  mean  of  the  times. 

However,  such  is  not  the  case  when  the  body  is  near  the 
meridian  in  azimuth,  at  which  time  the  changes  of  altitude 
are  proportional  to  the  squares  of  the  hour  angles. 

The  azimuth  tables  may  be  used  for  finding  the  time  when 
a  body  is  on  or  nearest  to  the  prime  vertical,  and  will  give 
results  sufficiently  accurate  for  all  practical  purposes. 

In  finding  the  t  corresponding  to  a  given  log  cos,  in  ex- 
amples 171  to  173,  when  cos  t  is  -{-,  use  the  p.  m.  column, 
dividing  through  by  2.  When  cos  t  is  — ,  use  the  a.  m. 
column,  adding  12  hours  to  the  reading  and  then  dividing 
by  2  to  obtain  t. 

For  a  body  whose  decimation  and  right  ascension  are 
changing  sufficiently  rapidly  to  require  correction  for  G.M.T., 
the  approximate  time  of  being  on  the  prime  vertical  must  be 
known  in  order  to  get  these  elements  for  use,  the  declination 
for  substitution  in  the  proper  formula,  and  the  right  ascen- 
sion to  be  applied  to  the  hour  angle  in  finding  the  L.  S.  T. 
to  be  converted  into  L.  M.  T.  To  get  the  approximate  time, 
when  the  body  is  on  the  prime  vertical,  assume  the  hour  angle 
or  take  it  from  the  azimuth  tables  and  apply  it  to  the  L.  M.  T. 
of  local  transit  from  the  Nautical  Almanac. 


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CHAPTEK  XVII. 
LATITUDE. 

240.  By  definition,  the  latitude  of  a  place  is  its  angular 
distance  North,  or  South  of  the  equator,  measured  on  the 
meridian  passing  through  the  place.  From  Art.  142  we 
know  that  latitude  is  the  declination  of  the  zenith  of  the 
place  or  the  altitude  of  the  elevated  pole  at  the  place,  and 
the  astronomical  work  of  finding  the  latitude  consists  in 
finding  one  or  the  other  of  these  arcs,  the  position  of  the  body 
determining  which  arc  should  be  found. 

In  working  for  latitude  the  elements  involved  are  Ji,  d,  and 
t,  and  the  relation  between  them  is  shown  in  the  fundamental 
trigonometric  equation, 

sin  h  =  sin  L  sin  d  +  cos  L  cos  d  cos  t. 

Now,  if  t  =  0,  as  when  the  heavenly  body  is  at  its  upper 
transit, 

sin  h  =  cos  z  =sin  L  sin  d  +  cos  L  cos  d, 
cos  z  =  cos  (L  —  d), 

z  =  L  —  d  and  L  —  z  -f-  d. 

L  =  z  +  d  is  the  general  formula  for  latitude  from  alti- 
tudes of  bodies  on  the  meridian.  The  value  of  L  depends  on 
the  values  of  z  and  d  and  the  method  of  their  combination. 
This  method  of  finding  latitude  by  observations  of  bodies  on 
the  meridian  is  the  simplest  as  well  as  the  most  accurate  one, 
the  results  being  independent  of  the  time  for.  all  practical 
purposes  (except  in  the  cases  of  the  moon  and  planets  where 
an  error  in  time  affects  the  declination),  and  having  no 


MERIDIAN  ALTITUDES 


489 


M, 


FIG.  114. 


greater  errors  due  to  altitude  than  the  error  in  the  altitude 
itself.  The  declinations  of  the  sun  and  stars  do  not  change 
rapidly,  hence  latitude  from  ob- 
servations of  these  bodies  is  but 
little  affected  by  errors  in  longi- 
tude or  time. 

When  a  heavenly  body  is  on 
the  upper  branch  of  a  meridian, 
its  declination,  zenith  distance, 
and  latitude  are  all  measured  by 
arcs  of  the  same  great  circle,  the 
proper  combination  of  any  two 
arcs  to  produce  the  third  being 
shown  in  the  illustrations  of  the  four  cases  considered. 

Let  Fig.  114  be  a  projection  of  the  celestial  sphere  on  the 
plane  of  the  meridian. 

NS  is  the  horizon ;  N  the  North,  8  the  South  point. 

Z  the  observer's  zenith. 

QQf  the  equator. 

PPr  the  axis  of  the  sphere. 

P  the  elevated  pole. 

P'  the  depressed  pole. 

QZ  the  declination  of  the  zenith  equals  the  latitude. 

NP  the  altitude  of  the  elevated  pole  equals  the  latitude. 

Let  M ! ,  M2 ,  M 3 ,  MI  be  the  four  positions  of  the  body  to 
illustrate  the  four  cases. 

(1)   Case  of  M!  whose  declination  is  of  a  different  name 
from  the  latitude,  or  negative. 

QM^  is  the  declination.    M^Z  is  the  zenith  distance 
=  90°  —  h  =  z. 

Then  QZ  =  M1Z  —  QM1, 

L  =  z  —  d.  (197) 


490  NAUTICAL  ASTRONOMY 

(2)   Case  of  M2  ,  decimation  +  and  <  L. 


L  —  z  +  d.  (198) 

(3)   Case  of  M3  ,  declination  +  and  >  L. 


L  =  d  —  z.  (199) 

In  all  these  cases  L  is  +,  d  is  -f-  or  (  —  )  as  it  is  of  the  same 
or  a  different  name  from  the  latitude.  In  (1)  and  (2)  d  is 
<  L  and  the  body  bears  towards  the  depressed  pole,  but  in 
both  cases  z  is  +  ;  therefore  it  is  to  be  marked  the  opposite 
of  the  bearing  of  the  body.  In  (3)  d  is  >  L  and  the  body 
bears  towards  the  elevated  pole,  but  in  the  formula  z  is  (  —  )  ; 
therefore  in  this  case  also  mark  z  the  opposite  of  the  body's 
bearing.  In  other  words,  give  d  its  proper  mark  N.  or  S.  If 
the  body  bears  North,  mark  z  S.;  if  it  bears  South,  mark  z 
N.  The  latitude  will  be  the  algebraic  sum,  with  the  name 
of  the  greater,  N.  or  S. 

(4)   Case  of  M±  ,  a  heavenly  body  at  its  lower  culmination. 
Then  'PN  —  M±N  +  PM4  =  M+N  +  (90°  —  Q'MJ 

L  =  h  +  p  =  h  +  90°  —  d.  (200) 

Formula  (199)  is  also  correct  for  this  case,  provided  we  use 
180°  --d  instead  of  d. 

Writing  (200)  thus,  h  =  L  —  p,  it  is  evident  that  the 
polar  distance  of  a  body  must  be  less  than  the  latitude  of  the 
place  in  order  that  the  body  may  be  visible  on  the  meridian 
below  the  pole.  A  body  visible  at  its  lower  transit  in  any 
latitude  is  termed  circumpolar  for  that  latitude.  The  sun's 
maximum  decimation  is  about  23£°,  or  the  polar  distance  a 
minimum  of  about  66  J°  from  the  North  pole  in  June,  or 
the  South  pole  in  December  ;  so  that  the  latitude  of  an  ob- 
server must  be  in  excess  of  about  66  J°  to  see  the  sun  at  its 
lower  transit,  in  North  latitude,  at  the  time  of  the  sun's 


MERIDIAN  ALTITUDES  491 

nearest  approach  to  the  North  pole,  as  in  June;  or  in  South 
latitude,  at  the  time  of  the  sun's  nearest  approach  to  the 
South  pole,  as  in  December.  On  the  supposition  that  alti- 
tudes under  10°  are  not  reliable,  owing  to  the  uncertainties 
of  refraction,  the  latitude  would- have  to  be  at  least  75°  and 
of  the  same  name  as  the  declination  to  justify  meridian  ob- 
servations of  the  sun  below  the  pole.  However,  stars  are 
available  in  all  latitudes  above  10°,  for  observations  under 
favorable  conditions  at  their  lower  transit,  and,  in  North 
latitude,  the  pole  star  is  available  at  all  times  when  visible 
and  if  of  sufficient  altitude. 

Since  a  heavenly  body  cannot  be  seen  at  its  lower  transit, 
unless  its  declination  is  positive,  or  of  the  same  name  as  the 
latitude,  it  follows  that  the  latitude  resulting  from  an  observa- 
tion of  a  body  crossing  the  lower  branch  of  the  meridian  will 
be  of  the  same  name  as  the  declination. 

From  the  formula  in  each  case  it  is  apparent  that  an  error 
in  a  meridian  altitude  produces  an  equal  error  in  the  result- 
ing latitude. 

In  all  meridian  observations  the  declination  of  the  heavenly 
body  must  be  taken  from  the  Nautical  Almanac  for  the  in- 
stant of  transit;  in  the  case  of  the  sun,  at  upper  transit  the 
declination  is  corrected  for  the  Greenwich  apparent  time  of 
noon,  which  in  West  longitude  equals  -f-  A  of  the  given 
date,  and  in  East  longitude  equals  —  X  of  the  given  date  or 
(24  hours  —  A)  of  the  day  before;  at  lower  transit  the  de- 
clination must  be  corrected  for  (12  hours  -f-  A)  if  A  is  West, 
or  for  (12  hours  —  A)  if  A  is  East,  in  either  case  for  the 
instant  of  local  apparent  midnight. 

In  the  case  of  stars,  the  declinations  do  not  change  with 
sufficient  rapidity  to  require  corrections,  and  hence  when  con- 
ditions are  favorable,  as  in  morning  or  evening  twilight,  ob- 
servations of  stars  on  or  near  the  meridian  are  desirable. 

In  the  case  of  the  moon,  the  time  should  be  accurately 


492  NAUTICAL  ASTRONOMY 

known,  owing  to  the  very  rapid  changes  of  declination;  for 
this  reason  the  moon  is  not  so  well  adapted  for  observation 
as  the  sun  or  stars. 

In  the  case  of  any  body  observed  on  the  meridian  for  lati- 
tude, the  sextant  altitude  must  be  reduced  to  the  true  alti- 
tude of  the  center  (Art.  212). 

Polaris  is  on  the  meridian  when  (Mizar)  £  Ursse  MajoriS; 
the  second  star  in  the  handle  of  the  "Dipper,"  is  vertically 
above  or  below  the  pole  star,  and  since  Polaris  changes  its 
altitude  very  slowly  when  crossing  the  meridian,  such  times 
are  the  best  for  observation  of  that  star.  However,  the  lati- 
tude may  be  found  without  appreciable  error  from  Polaris  at 
any  time  when  the  conditions  for  observation  are  favorable 
(see  Art.  254). 

241.  Work  preparatory  to  observing  a  meridian  altitude 
of  the  sun. — It  is  customary  at  sea  to  find  the  watch  time  of 
local  apparent  noon  (Art.  198),  to  begin  observations  10  to 
15  minutes  before  and  to  take  continuous  observations  till  the 
watch  shows  noon,  the  altitude  at  that  time  being  taken  as 
the  meridian  altitude.  However,  observations  should  be 
taken  till  the  sun  ceases  to  rise,  or  dips,  in  case  it  is  not  sta- 
tionary when  the  watch  indicates  noon,  the  maximum  alti- 
tude being  the  meridian  altitude,  subject  to  the  remarks  of 
Art.  246. 

Before  going  on  deck  the  navigator  should  have  found  the 
value  of  A0/i;  observing  the  sun  by  watch  at  15,  10,  or  a  few 
minutes  before  apparent  noon,  and  applying  the  correction  for 
A07t£2  (see  Art.  251),  he  knows  very  closely,  minutes  ahead  of 
time,  what  the  meridian  altitude  and  the  latitude  will  be.  Ho 
should  also  have  prepared  a  constant  which,  if  properly  ap- 
plied to  the  meridian  altitude  by  sextant,  will  give  the  lati- 
tude at  once.  The  latitude  and  longitude  being  approxi- 
mately known  by  D.  R.,  find  in  order  the  declination  and  the 
approximate  altitude,  for  which  take  out  the  parallax  in  alti- 
tude and  refraction. 


CONSTANT  FOR  LATITUDE  493 

Calling  the  algebraic  sum  of  the  I.  C.,  dip,  refraction, 
parallax,  and  semi-diameter  c,  the  sextant  meridian  altitude 
h8,  we  have: 

For  MI  ,  dec  (— ),  L  =  (90°  —  d  —  c)  —  hf. 

YorM2 ,  d  +  and  <  L,  L  —  (90°  +  d—  c)  —  h8. 
For  M3 ,  d  +  and  >  L,  L  =  h8—  (90°  —  d  —  c). 
For  M4  ,  lower  transit,  L  =  h8  -\-  ( 90°  —  d  +  c) 

or/i.+  (p  +  c). 

The  quantities  in  brackets  are  called  constants,  are  com- 
puted beforehand,  and  entered  in  the  navigator's  note-book. 
The  constant  is  applied  to  the  sextant  altitude  as  indicated 
for  each  particular  case. 

242.  To  find  the  latitude  from  the  sun's  meridian  altitude : 
(a)  at  upper  transit;  (b)  at  lower  transit. 

(a)  The  local  apparent  time  of  transit  is  Oh  Om  0s  of  the 
day  at  the  ship.     Find  the  G.  A.  T.  of  local  noon  by  apply- 
ing the  longitude ;  +  if  West,  —  if  East. 

(2)  Take  the  sun's  declination  from  page  I  of  the  Almanac 
for  the  proper  month  and  correct  it  for  the  G.  A.  T.,  marking 
it,  as  it  should  be,  N".  or  S.  (Art.  185). 

(3)  Eeduce  the  sextant  altitude  to  the  true  altitude  of  the 
center  (Art.  202).     Subtract  the  latter  from  90°  to  get  the 
zenith  distance.     If  the  body  bears  N.,  mark  the  zenith  dis- 
tance S.;  if  it  bears  S.,  mark  the  zenith  distance  N. 

(4)  The  latitude  is  the  sum  of  the  declination  and  zenith 
distance,  if  they  are  of  the  same  name,  and  is  marked  like 
them;  the  difference  of  the  two,  if  of  different  names  and 
marked  with  the  name  of  the  greater. 

(b)  At  lower  transit  the  local  apparent  time  is  12  hours. 
Find  the  corresponding  G.  A.  T.  and  for  it  the  declination  of 
the  sun,  then  the  polar  distance  =  90°  —  d. 

(2)  Reduce  the  sextant  altitude  as  above  explained. 

(3)  Add  the  altitude  and  polar  distance;  the  result  is  the 
latitude  marked  like  the  declination. 


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498 


NAUTICAL  ASTRONOMY 


243.  To  find  the  latitude  from  the  meridian  altitude  of  a 
fixed  star. 

Rules.  (1)  Find  by  computation  beforehand  the  local  and 
watch  time  of  transit  for  the  ship's  position  at  the  approxi- 
mate time  of  transit  (Arts.  196  and  198),  and  observe  at  that 
time.  Altitudes  should  be  obtained  before  and  after  the 
watch  time  of  transit  and  times  noted  for  use  in  case  the 
meridian  altitude  is  missed.  See  Art.  151  on  star  observations. 

Where  a  selection  can  be  made  about  a  given  time,  select  a 
star,  if  possible,  which  has  a  comparatively  low  altitude  for  ob- 
servation on  or  near  the  meridian,  because  such  a  star  will 
change  its  altitude  slowly  when  in  that  position. 

(2)  The  change  of  declination  of  a  fixed  star  is  so  slow  that 
it  may  be  neglected;  so  take  the  declination  direct  from  the 
apparent  place  table. 

(3)  Reduce  the  altitude  to  a  true  altitude  by  correcting  for 
I.  C.,  dip,  and  refraction,  unless  the  star  is  observed  with  an 
artificial  horizon,  in  which  case  proceed  as  in  Art.  154. 

(4)  Find  the  true  zenith  distance  and  combine  it  with  the 
declination  as  in  the  case  of  the  sun. 

Ex.  178. — At  sea,  January  21,  1905,  the  sextant  altitude 
of  the  star  a  Tauri  (Aldebaran)  on  the  meridian  bearing  S. 
was  69°  30'  10".  I.  C.  —  1'.  Height  of  eye  45  feet.  Find 
the  latitude. 


Altitudes,  &c. 

Altitude 
Corrections. 

jfc's  Declination. 

O      1      II 

#'s  sextant  alt.         693010   S 
Corr.  to  alt.                      7  58 

/    n 
T.  C.    -  1  00 
Dip     -  li  SB 
lief.  -  0  22 
Corr.  -  7  68 

*'s  Dec.  N  16°  18'  69". 

*'s  True  alt.              «9  22  12   S 
*'sTrue*                   20  37  48  N 
*'s  declination          1(5  18  59  N 
Latitude                     "38  66  47  N 

MERIDIAN  ALTITUDES 


499 


Ex.  179. — January  1,  1905,  the  sextant  altitude  of  the  star 
a  Eridani  (Achernar)  on  the  meridian  bearing  S.  was  35°  22'. 
I.  C.  +  2'.  Height  of  eye  20  feet.  Find  the  latitude. 


Altitudes,  &c. 

Altitude 
Corrections. 

#'s  Declination. 

rfc's  sextant  alt. 
Corr.  to  alt. 

o    /    n 
352200   S 
3  45 

/    it 
I.  C.   +2  00 
Dip    -  4  23 
Ref  .  -  1  22 

*'s  Dec.  S  67°  43'  32"  .6 

*'s  True  alt. 
*'s  True  z 
>K's  declination 

35  18  15  S 
54  41  45  N 
57  43  33  S 

Corr.  -  3  45 

Latitude 

3  01  48    S 

Ex.  ISO.— April  19,  1905,  the  sextant  altitude  of  the  star 
a  Aurigae  (Capella)  on  the  meridian  below  the  pole  was  11° 
20'.  I.  C.  +  2'.  Height  of  eye  40  feet.  Find  the  latitude. 


Altitudes, 

&c. 

Altitude 
Corrections. 

#'s  Dec.  and  Polar  Dist. 

^s  sextant  alt. 
Corr.  to  alt. 

o    /     tt 
11  20  00 
—       8  55 

i  it 
I.  C.    +  2  00 
Dip    —  6  12 
Ref.   -  4  43 

o    t        n 
*'sd=      N  45  54  04.3 
*'sp=           440565.7 

^:'s  True  alt. 
r^'s  polar  distance 

11  11  05 
44  05  56 

Corr.  —  8  65 

Latitude 

55  17  01  N 

The  latitude  has  the  same  name  as  the  star's  decimation, 
otherwise  the  star  would  not  be  visible  on  the  meridian  below 
the  pole. 

244.  To  find  the  latitude  by  the  meridian  altitude  of  the 
moon. — The  moon  being  comparatively  so  near  to  the  earth,  its 
changes  of  declination  and  semi-diameter  are  more  rapid  than 
in  the  case  of  other  bodies ;  besides,  its  parallax  is  quite  large. 
These  elements  require  careful  correction,  and  the  great  lia- 
bility to  error,  due  to  error  of  time,  render  observations  of  the 
moon  less  desirable  than  those  of  other  bodies.  When  the 
moon  is  near  the  equator  its  declination  changes  most  rapidly, 
and  at  such  times  the  maximum  altitude  may  differ  consider- 
ably from  the  meridian  altitude.  A  movement  of  the  zenith 


500  NAUTICAL  ASTRONOMY 

due  to  high  speed  of  observer's  ship,  especially  in  the  direc- 
tion of  the  meridian,  will  intensify  such  a  discrepancy  (Art. 
246)  ;  hence  it  is  better  to  calculate  the  time  of  meridian  pas- 
sage of  the  moon  and  consider  the  altitude  observed  at  that 
time  as  the  meridian  altitude. 

Rules.     (I)  Find  the  Greenwich  mean  time  and  date  of  the 
moon's  local  meridian  passage  (Art.  188). 

(2)  For  this  G.  M.  T.  find  the  moons  declination  S.  D. 
andH.P.  (Art.  185). 

(3)  Reduce  the  sextant  altitude  to  the  true  altitude  of  the 
center  (Art.  212). 

(4)  After  which.,  proceed  as  with  the  sun. 

245.  To  find  the  latitude  by  the  meridian  altitude  of  a 
planet. 

(1)  Find  from  the  Nautical  Almanac  the  G.  M.  T.  of 
Greenwich  transit.     To  this  apply  the  retardation  or  accelera- 
tion for  the  longitude,  and  the  result  will  be  the  L.  M.  T.  of 
local  transit  (Art.  189),  the  retardation  or  acceleration  per 
hour  being  one-twenty-fourth  of  the  difference  of  times  of 
transits  on  two  successive  days  as  indicated  in  the  Nautical 
A  Imanac. 

(2)  From  the  L.  M.  T.  of  local  transit  and  the  error  of  ob- 
server's watch  on  L.  M.  T.  find  the  watch  time  of  l^cal  transit 
and  observe  the  planet's  altitude  at  that  time.     Reduce  the 
sextant  altitude  to  a  true  altitude  by  applying  the  I.  C.,  dip, 
and  refraction,  neglecting  for  sea  observations  8.  D.  and  par- 
allax. 

(3)  To  the  L.  M.  T.  of  local  transit  apply  the  longitude 
and  obtain  the  G.  M.  T.  of  local  transit.     For  this  G.  M.  T. 
find  the  planet's  declination;  after  which>  proceed  as  in  the 
case  of  the  sun. 


MERIDIAN  ALTITUDES 


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MAXIMUM  ALTITUDES  503 

246.  Maximum  and  minimum  altitudes. — The  maximum 
altitude  of  a  heavenly  body  occurs  at  the  instant  of  upper 
transit,  provided  the  body  is  of  an  unchanging  declination, 
as  for  instance  a  fixed  star,  and  is  observed  from  a  fixed 
position.  However,  if  the  body  is  changing  its  declina- 
tion, and  if  the  observer  is  on  a  ship  in  motion,  changing 
his  horizon  and  zenith,  the  maximum  altitude  is  not  at  the 
instant  of  meridian  passage,  but  after,  if  the  body  and  zenith 
are  approaching,  before,  if  they  are  separating;  for,  if  ap- 
proaching, the  body  will  continue  to  rise  after  its  upper  tran- 
sit till  its  downward  velocity  equals  that  of  the  approach 
of  the  zenith  (or  observer).,  at  which  time  the  maximum  alti- 
tude is  reached;  if  the  body  and  zenith  are  separating,  the 
body  reaches  the  maximum  altitude  before  the  meridian  pas- 
sage and  at  a  time  when  the  velocity  of  rising  equals  that  of 
separation. 

Let  Ad  be  the  change  of  declination  in  1  minute,  if  ex- 
pressed in  seconds  of  arc ;  or,  in  1  hour,  if  expressed 
in  minutes  of  arc. 

Let  AL  be  the  change  of  the  zenith  with  respect  to  the  body 
in  the  meridian,  in  1  minute,  if  expressed  in  seconds 
of  arc;  or,  in  1  hour,  if  expressed  in  knots  or  min- 
utes of  arc  per  hour. 

Let  Ac  be  the  combined  action  of  Ad  and  AL,  causing  ap- 
proach or  separation,  expressed  in  the  same  units  as 
Ad  and  kL,  or  the  velocity  of  approach  or  separation. 
Let  A0fr  be  the  change  in  altitude  in  1  minute  from  me- 
ridian passage  due  solely  to  diurnal  rotation   (see 
Art.  251)  expressed  in  seconds  of  arc;  A7t,  the  cor- 
rection to  be  applied  to  the  maximum  altitude  to  re- 
duce it  to  the  meridian  altitude,  in  seconds  of  arc, 
sign  of  application  minus. 
Let  i  be  the  H.  A.  of  the  body  at  the  instant  of  highest 


504  NAUTICAL  ASTRONOMY 

altitude,  easterly  when  the  body  and  zenith  are  separating; 
westerly,  when  approaching.  It  will  be  apparent  or  sidereal 
time,  depending  upon  the  body  observed,  and  must  be  con- 
verted into  a  mean  time  interval  if  required  in  mean  time 
units. 

The  value  of  t  is  affected  by  the  change  of  longitude  made 
by  the  ship.  The  correct  H.  A.  is  obtained  by  increasing  an 
easterly  H.  A.,  or  decreasing  a  westerly  H.  A.,  for  a  westerly 
change  in  longitude,  and  the  reverse  for  an  easterly  change  in 
longitude. 
Then,  Ac  being  the  velocity  of  approach  or  separation, 

Ac£   will   be   the   change    in    altitude   produced   by 

changes  of  declination  and  latitude  in  the  time  t; 

A0ht2  (see  Art.  251),  the  diminution  of  altitude  due 

to  diurnal  rotation; 
and  A/i  =  Ac£  —  A07U2. 

Now,  since  the  velocity  of  the  body  on  the  meridian  is  zero, 
and  its  change  of  altitude  near  the  meridian  varies  as  t2  ,  its 
motion  near  the  meridian  is  uniformly  accelerated  or  retarded, 
according  as  t  is  +  or  (  —  )  . 

Therefore,  A07i£2  =  -J  at2  where  a  is  the  acceleration  or  re- 
tardation, and 

(201) 


From  the  formula  for  velocity  of  uniformly  accelerated 
bodies  V  •=.  v  +  a/,  we  have,  since  the  velocity  at  transit  is 
zero,  for  the  velocity  when  the  H.  A.  is  tf 

V  =  2A0R  (202) 

Now,  at  the  moment  of  maximum  altitude,  the  body  is  sta- 
tionary, its  velocity  in  altitude  equals  that  of  approach  or 
separation. 


MAXIMUM  ALTITUDE 


506 


Therefore,  2A07t£  =  Ac,  and  t  = 
and,  by  substitution, 


(203) 


At  the  lower  transit,  the  minimum  altitude  occurs  after 
the  meridian  passage,  if  the  body  and  zenith  are  separating; 
and  before,  if  approaching.  The  method  of  applying  the 
change  of  longitude  to  the  H.  A.  is  exactly  the  reverse  of  that 
at  the  upper  transit. 

In  view  of  the  many  inaccuracies  of  observations  at  sea,  this 
correction  is  not  of  any  importance  except  for  observations 
taken  on  board  very  swift  steamers  on  courses  near  the 
meridian. 

The  following  example  will  illustrate  the  preceding  re- 
marks : 

Ex.  184.— April  29,  1905,  in  latitude  50°  K,  longitude  25° 
30'  W.,  observed  from  a  ship  steaming  225°  (true)  20  knots 
per  hour  the  maximum  altitude  of  the  sun's  lower  limb 
to  be  54°  15'.  I.  C.  +  1'.  Height  of  eye  45  feet.  Cor- 
rected declination  of  the  sun  N.  14°  21'  34".  H.  D.  N.  46".83. 
Eequired  the  time,  a.  m.  or  p.  m.,  of  maximum  altitude,  and 
the  correction  to  reduce  this  altitude  to  the  meridian  altitude. 


Course. 


225 


Distance 


20 


14.1 


W 


14.1 


D  =  31'. 93  W 
4 

D  =  878.73  W 


Observer's  change  in  latitude  14M  S.  per  hour,  or  14". 1    S  per  minute, 
Change  in  the  sun's  declination  46". 83  per  hour,  or  0   .78  N       « 

Ac  =  combined  velocity  of  approach 14". 88  " 

For  Lat.  50°  N.  and  Dec.  14£°  N.  (Table  26,  Bowditch)  A0A  =  2".l 
Ac  14". 88         14". 88 


From  (203),  t 


_ 
2x2".l 


=  3»  32" 


As  observer's  zenith  and  body  approach,  I  is  a  westerly  H.A. 
As  the  ship  changes  her  longitude  to  the  westward,  at  the  rate 


506  NAUTICAL  ASTRONOMY 

of  87S.72  per  hour  or  5s  in  3m.54,  the  corrected  H.  A.  is 
_|_  3m  273^  an(}  the  time  of  maximum  altitude  is  3m  27s  after 
meridian  transit  or  3m  27s  p.  m. 

Aft  =  -*!*     =  25".  5. 


247.  Finding  latitude  by  observations  of  bodies  out  of  the 
meridian.  —  It  may  often  happen  that  the  sun  is  not  visible  at 
noon,  making  an  observation  for  latitude  desirable  at  its  ear- 
liest appearance  after  noon;  or  from  threatening  appearances 
in  the  morning,  the  indications  may  be  that  the  sun,  when  on 
the  meridian,  will  not  be  visible.  Under  such  circumstances 
forenoon  observations  should  be  taken.  In  fact  a  careful  navi- 
gator will  always  take  them  and  practically  know  his  noon 
position  ahead  of  time.  Besides,  during  morning  or  evening 
twilight,  or  moonlight,  there  are  many  stars  favorably  situ- 
ated and  available  for  latitude  observations;  so  methods  for 
finding  latitude  from  altitudes  of  bodies  out  of  the  meridian 
are  necessary. 

There  are  five  methods  in  use  : 

(1)  A  rigorous  method,  known  as  the  <£"<£'  method,  avail- 
able for  bodies  within  three  hours  of  the  meridian  and  whose 
azimuth,  ZN,  is  between  315°  and  45°  or  135°  and  225°,  and 
which  is  independent  of  latitude. 

(2)  An  approximate  method  involving  both  latitude  and 
longitude. 

(3)  By  reduction  to  the  meridian,  a  special  case  of  the 
second  method. 

(4)  By  altitude  of  the  pole  star,  using  modified  formulae 
of  the  first  method. 

(5)  Chauvenet's  method  by  two  altitudes  near  noon,  time 
unknown. 

(6)  Prestel's  method  by  rate  of  change  of  altitude  near 
the  prime  vertical  (only  an  approximation). 


EFFECT  OF  ERRORS  IN  DATA.  507 

248.  Effect  of  errors  in  data   on  the  latitude.— In  the 

solution  of  the  astronomical  triangle  for  latitude  from  an 
observation  of  a  heavenly  body  out  of  the  meridian,  all  the 
elements  involved,  except  the  declination,  may  be  consid- 
ered as  liable  to  error,  and  it  is  desired  to  find  the  effect 
on  the  latitude  of  (1)  an  error  in  altitude,  (2)  an  error  in  t, 
and  (3)  to  determine  the  most  favorable  position  of  a  heavenly 
body  for  observations  for  latitude. 

(1)  To  determine  the  effect  of  an  error  in  altitude  on  the 
resulting-  latitude. — Differentiating  the  fundamental  equation 

sin  Ti  =  sin  L  sin  d  -j-  cos  L  cos  d  cos  t, 
regarding  h  and  L  as  variables,  we  have 

cos  JidJi  =  sin  d  cos  LdL  —  cos  d  cos  t  sin  LdL 

dL_     cos  h 

dh  ~~  sin  d  cos  L  —  cos  d  cos  t  sin  L ; 
but,  from  trigonometry, 

sin  a  cos  B  =  cos  I  sin  c  —  sin  &  cos  c  cos  A, 
and  by  substitution, 

cos  h  cos  Z  =  sin  d  cos  L  —  cos  d  cos  t  sin  L. 
Therefore, 

cos  hdh  r, 

dL  — j- r  —  dh  sec  Z.  (204) 

COS  tl  COS  Zi 

(2)  To  determine  the  effect  of  an  error  in  t  on  the  lati- 
tude.— Differentiating  the  same  equation,  L  and  t  variables, 
we  have  from  equation  (189),  Art.  237, 

^t  =  cot  Z  sec  L;  therefore,  ^  =  cos  L  tan  Z  (dt  in  arc). 

Now,  since  dt  (in  arc)  equals  15dt  (in  time),  if  dt  is  given 
in  time, 

dL  =  15dt  cos  L  tan  Z.  (205) 

(204)  and  (205)  show  that  the  maximum  effect  of  errors 
in  altitude  and  time  are  produced  when  Z  =  90°,  the  mini- 


508 


NAUTICAL  ASTRONOMY 


mum  effect  when  Z  =  0°  or  180°,  the  inference  being  that 
positions  on  or  near  the  meridian  are  better  for  observations 
for  latitude,  and  that  observations  near  the  prime  vertical 
for  latitude  should  be  avoided. 

The  hour  angle  used  in  the  various  methods  is  very  liable 
to  error  at  sea,  either  from  error  in  the  original  determina- 
tion of  the  longitude,  or  error  in  run  to  time  of  observation 
for  latitude. 

If  the  sign  of  dL  due  to  dt  is  positive  when  the  body  is  on 
one  side  of  the  meridian,  it  will  be  negative  for  the  same  azi- 
muth on  the  other  side;  hence 
the  error  may  be  eliminated  by 
taking  the  mean  of  results  from 
observations  on  both  sides  of 
the  meridian. 

249.  To  find  latitude  by  an 
altitude  out  of  the  meridian. 
First  method.— To  find  the 
latitude  of  a  place  at  any  time, 
given  the  sextant  altitude  of  a 
heavenly  body,  the  Greenwich 
mean  time  of  observation,  and  the  longitude  of  the  place. 

Eeduce  the  sextant  altitude  to  the  true  altitude  of  the  cen- 
ter. Find  the  body's  declination  and  hour  angle  (Arts.  212 
and  185). 

Then  in  the  astronomical  triangle  PZM,  there  are  given 
ZPM  —  t,PM  —  90°  —  d,  ZM  =  90°  —  h,  and  it  is  re- 
quired to  find  PZ  =  90°  —  L. 

In  Fig.  115,  let  fall  from  M  a  perpendicular  Mm  on  the 
meridian.    Let  Pm  =  <f>  and  Zm  =  <£'. 
By  Napier's  rules,  we  have 

tan  <£  =  cot  d  cos  t, 

cos  <j>'  =  cos  <f>  sin  h  cosec  d,  \-  (206) 

L  =  90°—  (<#>±  </>')• 


FIG.  115. 


'  METHOD 


509 


(207) 


Following  Chauvenet's  methods,  the  above  can  be  put  into 

more  convenient  form. 

If  for  <£  in  the  above,  90°  —  <j>"  be  substituted,  then 

tan  <£"  =  tan  d  sec  t, 

cos  <£'  =  sin  </>"  sin  h  cosec  d, 
L  =  </>"  zp  <£'. 

Case  of  M ,  d  +  and  >  L. 
From  the  figure  it  is  seen  that 

$    —  Pm,  the  polar  distance  of  m,  the  foot  of  the  per- 
pendicular ; 

<£"  —  Qm,  the  declination  of  m,  the  foot  of  the  perpen- 
dicular ; 

<£'  —  mZ,  the  zenith  distance  of  m,  the  foot  of  the  per- 
pendicular ; 
L  =  QZ,  the  declination  of  the  zenith. 

But  QZ  —  Qm  —  mZ  or  L  =  <j>"  —  <£'. 


FIG.  116. 


As  shown  in  the  figure  (115),  the  declination  of  M  is  posi- 
tive and  >  L,  and  finding  the  latitude  in  this  case  resolves 
itself  into  the  finding  of  the  declination  and  zenith  distance 
on  the  meridian  of  the  foot  of  the  perpendicular  and  combin- 


510  NAUTICAL  ASTRONOMY 

ing  them  by  the  rules  applying  to  a  similar  case  in  finding 
latitude  from  the  meridian  altitude  of  a  body. 
Case  of  M  2  ,  d  -f-  and  <  L. 

In  triangle  PZM2  (Fig.  116),  <£"  =  Qm2  and  </>'  =  mzZ. 
QZ  =  Qm2  +  m2Z  or  L  =  <£"  +  <£'. 

Case  of  If3  ,  d  is  negative. 

In  triangle  PZM3  (Fig.  116),  0"  =  Qm3  and  0'  =  w3^. 
QZ  =  m3Z  —  Qm3  or  L  =  <j>f  —  <j>". 

Case  of  M^  ,  t  >  6  hours,  J  +. 
In  triangle  PZM4,  <£"  =  Qm±  and  <£'  =  m^Z. 
QZ  —  Qm±  —  m±Z  =  </>"  —  </>'. 

In  this  case  of  M  4  ,  t  is  >  6  hours  and  d  is  +  ;  therefore, 
Qm^  is  taken  out  +,  same  sign  or  name  as  d,  but  in  the  second 
quadrant  —  or  same  quadrant  as  t. 

Now,          0ra4  =  </>"  =  90°  +  Pm^  =  90°  +  p, 
m±Z  =  <!>'  =  90°  —Nm4  =  90°  —  7t, 

~~li  =  L. 


Therefore  this  case  corresponds  to  that  of  a  body  observed  on 
the  meridian  below  the  pole. 

As  <£'  is  found  from  its  cosine  it  may  be  either  -f-  or  —  ,  thus 
giving  two  values  of  L,  differing  largely  from  each  other,  un- 
less </>'  is  small.  However,  the  latitude  is  approximately 
known,  and  no  trouble  need  be  experienced  in  determining 
how  to  mark  cf>'. 

The  following  rules,  closely  attended  to,  will  prevent  any 
error  in  the  proper  marking,  or  the  method  of  combining 
<£"  and  </>'  to  obtain  the  latitude. 

(1)  The  mere  fact  of  t  being  +.  (W.)f  or  —  (E>),  has  no 
influence  on  the  signs  of  the  functions.  If  t  is  >  6  hours 
sec  t  is  (  —  )  and  <j>"  is  in  the  second  quadrant.  Therefore, 


(/>"  </>'  ^i  i  TII oi)  511 

(2)  In  formula  (207)  <j>"  is  taken  out  in  the  same  quad- 
rant as  t  and  is  marked  N.  or  S.  like  the  declination.  <j>'f 
being  the  zenith  distance  of  m,  is  marked  like  the  zenith  dis- 
tance of  the  body  in  a  meridian  observation  for  latitude;  that 
is,  if  the  body  bears  northerly,  mark  the  zenith  distance  8.,  if 
it  bears  southerly,  mark  the  zenith  distance  N.  Then  combine 
</>"  and  <j>f  algebraically  according  to  their  names.  The  result 
will  be  the  latitude. 

Under  the  following  conditions  this  method  is  not  condu- 
cive to  accuracy,  or  fails  entirely. 

(1)  When  </>'  is  very  small,  that  is,  when  Z  is  near  90°  and 
the  body  is  near  the  prime  vertical,  it  cannot  be  found  accu- 
rately from  the  cosine. 

(2)  When  d  is  0,  <£"  is  0,  <£'  is  indeterminate,  and  the  lati- 
tude cannot  be  found  by  this  method. 

Observations  of  the  sun,  planets,  or  fixed  stars,  worked  for 
latitude  by  this  method  give  excellent  results;  owing  to  the 
very  rapid  changes  of  the  moon's  elements,  and  the  uncer- 
tainties of  the  hour  angle  due  to  the  uncertainties  of  longi- 
tude, observations  of  the  moon  are  not  recommended. 

Rules  for  Working  a  </>"<£'  Sight. 

(1)  Find  the  G.  M.  T.  of  observation  for  which  in  the  case 
of  the  sun,  take  from  the  Nautical  Almanac  the  declination 
and  equation  of  time;  or,  in  the  case  of  any  other  body,  its 
right  ascension  and  declination  and  the  right  ascension  of  the 
mean  sun,  and  also,  if  the  moon  has  been  observed,  its  semi- 
diameter  and  horizontal  parallax;  and  reduce  the  sextant  alti- 
tude to  the  true  altitude  of  the  center. 

(2)  Find  the  body's  t,  then  having  t,  d,  and  h,  proceed  by 
substitution  in  formula  (207)  to  find  $"  and  <f>',  paying  par- 
ticular attention  to  the  rules  preceding  regarding  the  sign  of  t, 
and  the  naming  and  combining  <£"  and  <£'  to  obtain  the  lati- 
tude. 


512 


NAUTICAL  ASTRONOMY 


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514  NAUTICAL  ASTRONOMY 

250,  Second  method. — When  the  latitude  is  approximately 
known. 

Taking  the  fundamental  equation 

sin  Ji  =  sin  L  sin  d  -f-  cos  L  cos  d  cos  t, 

and,  substituting  for  cos  t  its  equivalent  1 — 2  sin2  -J  i,  we  have 

sin  h  =  sin  L  sin  d  +  cos  L  cos  d  —  2  cos  L  cos  d  sin2  £  £ 

—  cos  (L  ~  d)  —  2  cos  L  cos  d  sin2  J  tf, 

but  (L  ~  d)  =  z0  =  90°  -  ^0  where  Ji0  represents  the 
meridian  altitude  of  the  body  at  some  place  in  the  same  lati- 
tude as  the  observer  at  the  same  instant  when  the  body's 
declination  is  d. 

Therefore,  sin  Ji  =  sin  h0  —  2  cos  L  cos  d  sin2  £  t 

and  sin  Ji0  =  sin  li  +  2  cos  L  cos  d  sin2  J  £.       (208) 

The  approximate  latitude  is  used  in  finding  the  value  of 
2  cos  L  cos  d  sin2  £  ^  and  then  from  the  formula  an  approxi- 
mate value  of  the  meridian  altitude  is  computed.  Having 
found  the  declination  for  the  Greenwich  instant  of  observa- 
tion, the  latitude  is  then  found  as  in  the  case  of  a  meridian 
observation  (Art.  240). 

The  nearer  the  body  is  to  the  meridian  at  the  time  of  obser- 
vation, the  more  correct  will  be  the  resulting  latitude.  Ob- 
serving at  equal  altitudes  on  opposite  sides  of  the  meridian 
will  eliminate  effect  of  errors  in  time. 

If  the  computed  latitude  differs  largely  from  the  assumed 
approximate  latitude,  repeat  the  computation,  using  in  the 
formula  the  computed  latitude.  It  is  seldom  necessary  to  re- 
peat the  computation  more  than  once.  In  the  example  that 
follows,  30'  more  of  assumed  latitude  would  have  increased 
the  computed  latitude  only  27". 

Eeference  has  been  made  in  Art.  219  to  the  use  of  a  for- 
mula involving  haversines  in  finding  Z,  and  in  Art.  226  to  a 
similar  formula  for  finding  t  and  thence  the  longitude.  For 


BY  SINE  FORMULA 


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516  NAUTICAL  ASTRONOMY 

a  navigator  who  may  have  Inman's  Tables,  the  following  for- 
mula for  latitude  is  recommended  as  being  simpler  than  the 
one  above  used,  by  one  step  in  the  calculation. 
Taking  the  fundamental  formula, 

sin  In,  =.  sin  L  sin  d  +  cos  L  cos  d  cos  t 
and  substituting  for  cos  t  its  equivalent  1  —  versin  t,  and  for 
sin  li,  its  equivalent  cos  z,  we  have 

cos  z  =  cos  (L  ~  d)  —  cos  L  cos  d  versin  t; 

but  cos  (L  ~  d)   =  cos  ZQ  when  ZQ  is  the  meridian  zenith 
distance;  therefore, 

cos  z  =  cos  z0  —  cos  L  cos  d  versin  t. 
Taking  each  side  from  unity, 

1  —  cos  z  =  1  —  cos  z0  +  cos  L  cos  d  versin  t, 

or  versin  z  =  versin  z0  +  cos  L  cos  d  versin  t, 

versin  z0  =  versin  z  —  cos  L  cos  d  versin  t . 

Now  if  we  let  versin  6  =  cos  L  cos  d  versin  i, 

then,  haver  0  =  cos  L  cos  d  haver  t  (to  determine  6),     (209) 

and  haver  z0  =  haver  z  —  haver  6.  (210) 

Having  found  the  meridian  zenith  distance,  proceed  as  above. 

251.  Third  method. — Reduction  to  the  meridian. — When 
an  observation  of  a  heavenly  body  is  taken  very  near  the 
meridian,  and  t,  the  hour  angle  of  the  body  East  or  West, 
is  known,  the  observed  altitude  may  be  reduced  to  what 
would  be  the  meridian  altitude  at  the  same  place  by  apply- 
ing a  correction  called  "The  Eeduction  to  the  Meridian," 
the  declination  of  the  body  being  assumed  the  same  at  the 
time  of  observation  and  when  on  the  meridian.  In  the 
American  naval  service  the  method  is  known  as  that  of  "  The 
Reduction  to  the  Meridian";  in  the  British  navy  as  that  of 
"  The  Ex-Meridian." 


HAVERSINE  METHOD 


517 


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do 


518  NAUTICAL  ASTRONOMY 

To  find  the  Reduction  to  the  Meridian. 

From  (208) 

^in  h0  =  sin  h  -\-  2  cos  L  cos  d  sin2  -J  i, 
or  sin  h0  —  sin  h  =  2  cos  L  cos  d  sin2  J  t. 
From  trigonometry, 

sin  x  —  sin  y  =  2  cos  -J  (a?  +  y)  sin  ^  (#  —  ^). 
Therefore, 

cos  |  (7i0  +  h)  sin  -J  (7i0  —  7i)  =  cos  L  cos  d  sin2  J  1 
Now  7i0  and  7&  differ  but  little;  therefore,  cos  \  (hQ  -(-  Ji)  may 
be  considered  as  cos  h0  =  sin  (L  ~  d). 

Letting  ATt  be  7z,0  —  7i,  which  is  the  "  Eeduction  to  the  Me- 
ridian "  desired,  we  have,  by  substitution, 

sin  J  ATfc  =  cos  L  cos  d  sin2  J  t  cosec  (L  ~  d). 

Since  the  body  is  near  the  meridian,  A7&  and  A£  are  assumed 
to  be  small,  and  we  have 

sin  J  Aft  —  -J  ATi"  sin  lr/  (ATi  expressed  in  seconds  of  arc), 
sin  J  £     =  -J  (15£)  X  sin  1"  (  ^expressed  in  seconds  of  time) ; 
and,  by  substitution  in  the  preceding  equation, 

Aft"  =  112.5  t2  cos  L  cos  d  cosec  (L  ~  d)  sin  1". 

In  the  above,  t  is  in  seconds  of  time,  but  if  we  wish  the 
hour  angle  to  be  expressed  in  minutes  of  time,  we  must  sub- 
stitute 60 1  for  t  in  the  equation. 
Therefore, 

A7i"  =  112.5  (6002  X  .000004848  cos  L  cos  d  cosec  (L  ~  d). 
A7i"  =  1".96349  t2  cos  L  cos  d  cosec  (L~d).  (211) 

If  t  is  one  minute, 

A07t"  =  1".96349  cos  L  cos  d  cosec  (L~d).         (212) 

Then  the  meridian  altitude  is 

h0  =  h  +  ATz,"  =  ft  +  A07z/'/2,  (213) 


EEDUCTION  TO  MERIDIAN  519 

in  which  h0  is  the  meridian  altitude  of  the  center  of  the  body ; 
h,  the  true  altitude  at  observation ; 
A/z/',  the  "  Eeduction  to  the  Meridian  "  in  seconds  of 

arc; 
A0&,  the  change  of  altitude  expressed  in  seconds  of 

arc  for  one  minute  of  time  from  the  meridian; 
t,  the  body's  hour  angle  expressed  in  units  of  its  own 

time. 

The  sign  of  application  of  Aft  is  always  positive  to  an  ob- 
served altitude  near  upper  transit,  negative  to  one  near  lower 
transit. 

To  find  the  Hour  Angle  t. 

Find 'the  watch  time  of  the  body's  transit  (Art.  198) ;  sub- 
tracting from  this  the  watch  time  of  observation,  the  result 
will  be  the  mean  time  interval  between  transit  and  observa- 
tion. The  difference  between  apparent  and  mean  time  inter- 
vals is  so  small  that  the  mean  time  interval  may  be  taken  as 
the  sun's  hour  angle  without  correction.  As  a  fixed  starts 
hour  angle  is  expressed  in.  sidereal  units,  in  the  case  of  a  star 
observation,  the  above  mean  time  interval  must  be  converted 
to  a  sidereal  time  interval  to  give  the  star's  hour  angle  at  the 
time  of  observation. 

In  the  case  of  a  planet,  we  may  disregard  the  slight  change 
in  right  ascension  and  take  the  sidereal  interval  as  its  hour 
angle. 

In  the  case  of  the  moon,  owing  to  the  rapid  change  of  its 
right  ascension,  the  above  method  will  not  do,  and  the  hour 
angle  must  be  found  in  lunar  units.  To  do  this,  find  the 
G-.  M.  T.,  then  the  L.  S.  T.,  corresponding  to  the  watch  time 
of  observation  (see  Art.  199  (b));  the  difference  between 
this  L.  S.  T.  and  the  moon's  right  ascension  for  the  Green- 
wich instant  of  observation  will  be  the  required  hour  angle  of 


520  NAUTICAL  ASTRONOMY 

the  moon.  However,  owing  to  the  rapid  changes  of  the 
moon's  right  ascension  and  declination,  observations  of  the 
moon  for  reduction  to  the  meridian  are  not  recommended. 

To  find  A0ft. 

It  may  be  found  in  Table  26,  Bowditch,  where  it  is  tabu- 
lated for  each  degree  of  latitude  from  0°  to  60°  and  each 
degree  of  declination  from  0°  to  63°,  there  being  a  table  for 
the  case  when  the  declinations  are  all  of  the  same  name,  also 
of  a  different  name  from  the  latitude.  No  values  are  given 
when  L  ^  d  is  <  4°,  or,  in  other  words,  when  the  altitudes 
are  above  86°,  as  the  method  is  inapplicable  when  the  body 
transits  so  near  the  zenith.  Furthermore,  no  values  are  given 
in  those  cases  where  a  body's  altitude  would  be  less  than  6°,  as 
such  altitudes  themselves,  owing  to  the  uncertainties  of  re- 
fraction, are  unreliable. 

It  may  also  be  found  in  Table  III  of  the  Ex-Meridian 
Tables  of  Brent,  Walter,  and  Williams,  under  the  designa- 
tion "  C." 

Table  26  gives  A07z,  to  the  nearest  tenth  of  a  second  of  arc 
only;  if  a  closer  approximation  is  desired,  A0fe  must  be  com- 
puted by  the  formula 

A07z,  =  1".96349  cos  L  cos  d  cosec  (L  ~  d). 
If  d  is  of  a  different  name  from  L.,  (L  ^  d)  becomes  numer- 
ically the  sum  of  the  two. 

If  A0/i  is  desired  for  any  case  not  tabulated,  it  may  be  com- 
puted by  the  above  formula. 

To  find  Aft. 

In  (211),  A/i"  is  in  seconds  of  arc  and  represents  the  change 
in  the  altitude  near  the  meridian  for  t  minutes  of  hour  angle 
expressed  in  time. 

The  value  of  ATz,  expressed  in  minutes  and  seconds  of  arc 
may  be  found  from  Table  27,  Bowditch,  the  arguments  being 


KEDUCTION  TO  MERIDIAN  521 

t  (in  minutes  and  seconds  of  time)  and  A0/i  (in  seconds  of 
arc).  The  larger  the  value  of  A0/i,  the  smaller  is  the  limit 
of  t;  thus  for  a  value  of  A07t  =  2".0,  AA,  is  given  for  t  as 
great  as  26  minutes,  while  for  a  value  of  A07*,  —  28",  it  is 
given  only  for  values  of  t  of  and  less  than  5  minutes.  It  may 
also  be  found  from  Table  IV  of  Brent,  Walter,  and  Williams' 
Ex-Meridian  Tables. 

Limit  of  H.  A. — The  following  most  excellent  rule  is  given 
in  "  Wrinkles  in  Navigation  " :  "  The  hour  angle  of  the  sun 
or  time  from  noon  in  minutes  should  not  exceed  the  number 
of  degrees  in  the  sun's  meridian  zenith  distance."  This  rule 
is  made  general  by  saying,  "  The  hour  angle  in  minutes  from 
the  meridian  of  a  body  observed  for  latitude  by  reduction  to 
the  meridian  should  not  exceed  the  number  of  degrees  in  its 
meridian  zenith  distance."  Chauvenet  has  shown  that  so 
long  as  the  zenith  distance  is  not  greater  than  10°,  the  re- 
duction computed  as  above  may  amount  to  as  much  as  4?  30" 
without  being  in  error  more  than  1".  The  rule  should  not  be 
made  applicable  to  circumpolar  stars  in  whose  cases,  the  limits 
of  H.  A.  may  be  greatly  extended. 

As  a  practical  rule,  in  all  cases  when  the  time  from  the 
meridian  transit  exceeds  the  limit  laid  down  in  Table  27, 
Bowditch,  it  would  be  better  to  find  the  latitude  by  the  <f>"<f>' 
method. 

Limits  of  Table  27. — The  limits  of  this  table  are  the  limits 
within  which  the  method  may  be  used  with  a  fair  degree  of 
confidence  in  the  accuracy  of  results. 

The  use  of  the  table  depends  on  the  object  sought;  for  in- 
stance, in  determining  latitude,  when  surveying,  the  hour 
angles  of  bodies  observed  should  be  so  small  that  the  value  of 
A/&  itself  should  not  exceed  1' ;  on  the  high  seas,  the  reduction 
obtained  under  conditions  in  which  even  the  limits  of  the  table 
are  used  will  be  sufficiently  exact.'  Again  the  table  may  be 
used  at  sea  beyond  its  limits  in  the  following  way,  if  this  use 


522  NAUTICAL  ASTRONOMY 

is  desired:  Suppose  A0ft  =  4".0  and  t  =  24  minutes;  since 
the  table  does  not  give  Aft  for  t  beyond  21  minutes,  find  the 
reduction  for  12  minutes  and  multiply  it  by  4  to  obtain  Aft 
for  24  minutes.  Therefore,  the  required  Aft  =  (9'  36")  X  4 
=  38'  24".  Proof  of  this  is  seen  from  the  fact  that 

4. 

Hence,  if  the  hour  angle  is  greater  than  the  tabulated  one 
for  the  given  value  of  A0ft,  take  out  the  correction  Aft  for  one- 
half  the  H.  A.  and  multiply  it  ~by  4;  the  result  will  be  the 
required  Aft. 

Eestrictions  in  the  tropics. — In  the  tropics,  where  at  transit 
the  body's  altitude  may  approach  90°,  the  factor 

cosec  (L  ~  d) 

will  be  so  large  as  to  make  Aft  too  great  for  the  assumption 
made  in  the  deduction,  that  sin  J  Aft  =  £  Aft  sin  1"  For 
such  cases  the  value  of  A0ft  is  not  tabulated.  In  those  regions, 
therefore,  in  summer  time,  this  method  is  not  applicable; 
however,  it  is  not  much  needed  owing  to  the  strong  probability 
of  the  sun  being  visible  when  on  the  meridian. 

To  Find  the  Declination  and  Latitude. 

The  declination  to  be  combined  with  the  meridian  alti- 
tude ft0  should  properly  be  corrected  for  the  Gr.  M.  T.  of  ob- 
servation ;  in  the  case  of  the  moon  this  is  essential ;  in  the  cases 
of  a  planet  or  the  sun,  it  is  sufficiently  accurate  to  use  the  de- 
clination at  the  instant  of  meridian  transit,  except  when  the 
hour  angle  is  large,  and  in  the  case  of  the  sun,  therefore,  the 
declination  may  be  corrected  for  the  longitude  at  upper  tran- 
sit and  for  (12  hours  -f  A)  at  the  lower  transit. 

Having  found  ft0  and  df  the  latitude  is  found  as  in  the  case 
of  a  meridian  altitude  (Art.  240). 

It  must  not  be  forgotten  that  the  latitude  thus  found  is  for 
the  instant  of  observation,  and  that  the  latitude  at  the  time 


"REDUCTION"  TABLES  523 

of  transit  (or  in  the  case  of  a  sun  sight,  near  the  upper  merid- 
ian, the  latitude  at  noon)  may  be  found  by  applying  the  run 
in  the  interval. 

Errors  in  H.  A. — The  effect  of  errors  in  H.  A.  may  be  min- 
imized by  observing  the  body  at  practically  the  same  altitude 
and  with  small  hour  angles  on  opposite  sides  of  the  meridian, 
reducing  the  latitudes  found  to  noon  and  taking  the  mean  of 
results  for  the  true  noon  latitude. 

Various  "Reduction"  or  "Ex-Meridian  Tables."— The 
Tables  of  Bowditch  and  of  Brent,  Walter,  and  Williams,  which 
are  practically  identical,  have  been  referred  to.  The  argu- 
ments in  these  Tables  are  L,  d,  and  t;  so  the  navigator,  having 
set  his  watch  to  L.  A.  T.,  may  have  in  his  note-book  the  cor- 
rections to  be  applied  to  altitudes  to  be  observed  at  certain 
times  by  the  watch  to  obtain  the  meridian  altitudes,  in  fact 
have  ready  a  constant  allowing  for  the  run  to  noon  (see  Art. 
253),  so  that  the  noon  latitude  may  be  found  at  once  by  apply- 
ing this  constant  to  the  observed  altitude.  Besides,  the  above- 
mentioned  tables  are  applicable  to  bodies  of  a  declination  as 
large  as  63°. 

Towson's  Tables  are  also  issued  to  the  American  navy. 
They  are  not  applicable  to  bodies  whose  declination  may  be 
greater  than  23°  20'.  The  arguments  used  in  these  tables 
are  d,  h,  and  t,  so  that  the  correction  is  taken  out  after  the 
altitude  has  been  observed;  a  matter  of  delay  if  not  of  in- 
convenience. 

There  have  been  various  graphic  and  automatic  methods 
for  finding  the  value  of  the  "  Reduction  to  the  Meridian,"  the 
best  of  which  perhaps  is  the  invention  of  Wm.  Hall,  Naval 
Instructor,  R.  N.,  and  which  consists  of  two  calculating  slides 
for  automatic  calculation.  It  is  known  as  "Hall's  Nautical 
Slide  Rule/' 

Rules. — (1)  Find  the  watch  time  of  transit  (Art.  198), 
and  the  H.  A.  from  the  meridian,  remembering  it  is  to  be  in 
sidereal  time  for  a  fixed  star,  and  that  for  the  sun  the 


524  NAUTICAL  ASTRONOMY 

mean  time  interval  may  be  used.  (2)Take  from  the 
Nautical  Almanac  the  declination  for  the  instant  of  transit,, 
or  in  the  case  of  the  sun,  for  local  apparent  noon  (if  sun  was 
observed  near  lower  transit,  for  local  apparent  midnight). 
(3)  From  Table  26,  Bowditcli,  take  out  A0/t  and  from  Table 
27,  A/i.  (4)  Reduce  the  sextant  altitude  to  a  true  altitude  of 
the  center  and  apply  £Ji;  adding,  when  the  body  was  observed 
near  upper  transit,  subtracting,  for  an  altitude  near  lower 
transit.  The  result  will  be  the  meridian  altitude.  (5)  Then 
proceed  to  find  latitude  as  in  Art.  240. 

Attention  is  called  to  the  fact  that  formula  (212)  is  made 
applicable  to  the  case  of  a  body  near  lower  transit  either  by 
substituting  180°  —  d  for  d,  or,  by  substituting  — L  for  L 
since  the  lower  transit  of  the  meridian  in  a  given  latitude  is 
the  upper  transit  of  the  same  meridian  at  the  antipode;  hence 
for  a  body  below  the  pole  take  A0/i  from  that  part  of  Table  26 
in  which  L  and  d  are  of  different  names  (see  bottom  of  last 
three  pages  of  Table  26). 


KEDUCTION  TO  MERIDIAN 


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NAUTICAL  ASTRONOMY 


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EEDUCTION  TO  MERIDIAN 


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528  NAUTICAL  ASTRONOMY 

252.  To  find  the  latitude  from  a  number  of  altitudes  of  a 
heavenly  body  observed  very  near  the  meridian,  the  longi- 
tude and  Greenwich  times  being  known.  —  Very  near  the 
meridian,  the  change  of  altitude  varies  nearly  as  the  square 
of  the  hour  angle,  so  that  the  mean  of  the  altitudes  cannot  be 
taken  as  corresponding  to  the  mean  of  the  times,  but  each 
altitude  may  be  reduced  to  the  meridian  by  the  principles  of 
Art.  251,  and  the  mean  of  these  used  in  finding  the  latitude, 
hence  the  term 

Circummeridian  Altitudes. 

Let  7&±  ,  h2  ,Jiz  ......  Jin  be  the  several  true  altitudes  ; 

/!  ,  t2  ,  tj  .......  tn  be  the  corresponding  hour  angles  in 

minutes  of  time  at  the  times  of 
observations  ; 
&Ji,  A27z,,  A37t  .  .  .  Anft  be  the  several  reductions  to  the 

meridian  ; 

where  A^  =  &oht\  ,  A2fe  =  A0/^22,  and  An/t  =  A0/i£2n  . 
Then  for  n  observations,  the  mean  value  of  the  meridian  alti- 
tude will  be 


n  n 

Substituting  the  values  of  the  reductions  as  above, 


,+ft,  ......  kn+  (f.  +<•.+*.  .....  <•.)  ^        (2U) 

From  this  value  of  h0  ,  the  meridian  altitude,  the  latitude  is 
computed.  The  principles  involved  in  this  method  suppose 
the  declination  not  to  change  from  the  time  of  observation 
till  the  meridian  passage,  and  as  the  declination  for  that  in- 
stant is  wanted  to  combine  with  what  would  be  the  meridian 
altitude,  it  is  better  to  find  the  Greenwich  time  of  passage 
(see  Art.  196),  and  for  this  take  out  the  body's  declination. 
Then  find  the  watch  time  of  passage  (Art.  198),  the  differ- 
ences between  which  and  the  watch  times  of  observation  will 


ClECUMMERIDIAN    ALTITUDES  529 

be  the  hour  angles  of  the  body  expressed  in  mean  time.  The 
mean  time  interval  differs  from  the  apparent  time  interval 
only  by  the  change  in  the  equation  of  time  in  the  interval ;  so 
in  the  case  of  the  sun  the  mean  time  intervals  need  not  be 
reduced. 

In  the  case  of  a  fixed  star,  they  must  be  converted  into  sid- 
ereal time  intervals. 

The  values  of  refraction  and  parallax  for  the  various  alti- 
tudes will  differ  so  slightly,  that  it  will  be  sufficient  to  re- 
duce the  mean  of  the  sextant  altitudes  to  a  true  altitude,  to 
which  the  reduction  will  be  applied  to  give  the  true  meridian 
altitude. 

When  possible,  altitudes  at  about  the  same  hour  angles 
should  be  taken  on  both  sides  of  the  meridian  in  order  to 
eliminate  errors  due  to  the  time. 

The  best  results  are  gotten  when  two  stars  culminating  at 
about  the  same  altitude  are  observed  on  opposite  sides  of  the 
zenith;  for,  by  taking  a  mean  of  the  two  latitudes  thus  ob- 
tained, personal  and  instrumental  errors,  if  the  instruments 
are  used  in  the  same  way  and  under  like  conditions,  are  elimi- 
nated. In  using  this  method  on  shore,  if  prismatic  effect  is 
suspected  in  the  roof  of  the  artificial  horizon,  it  would  be 
better  to  take  two  sets  of  observations,  the  roof  being  reversed 
between  the  sets. 

At  sea,  single  observations  near  the  meridian  are  sufficient 
and  A0&  from  the  tables  are  accurate  enough;  but  for  refined 
determinations  of  latitude  on  land,  it  is  better  to  take  a  num- 
ber of  observations  near  transit,  on  both  sides  of  the  meridian, 
using  a  bright  star  in  preference  to  the  sun,  and  computing 
the  value  of  A07L 

The  barometer  and  thermometer  should  be  noted  during 
observations  ashore,  and  a  correction  (Tables  21  and  22,  Bow- 
ditch),  dependent  upon  the  instrumental  indications,  should 
be  applied  to  the  mean  refraction. 


530 


NAUTICAL  ASTRONOMY 


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532  NAUTICAL  ASTRONOMY 

253.  To  find  a  constant  for  latitude  by  circummeridian 
altitudes  of  the  sun  near  upper  transit.  Before  going  on 
deck  for  the  meridian  observations,  the  navigator  should  pre- 
viously have  obtained  the  following  data  for  use  in  determin- 
ing the  noon  position;  viz.,  the  longitude  at  noon  correspond- 
ing to  a  given  latitude  and  the  change  in  longitude  for  1'  of 
latitude;  so  that,  by  a  slight  mental  calculation,  he  can  obtain 
the  noon  longitude  as  soon  as  he  has  determined  the  true 
noon  latitude  (see  Art.  301)  and  be  able  to  report  both  lati- 
tude and  longitude  wnen  he  reports  twelve  o'clock. 

To  obtain  the  latitude  ahead  of  time,  he  should  know  not 
only  the  constant  for  latitude  by  meridian  altitude  (Art.  240), 
but  the  constants  which  will  give  the  noon  latitude,  if  properly 
applied  to  sextant  altitudes  of  the  sun  at  given  hour  angles 
from  noon;  these  are  gotten  from  the  former  by  applying  a 
correction  consisting  of  two  parts:  first,  a  reduction  to  the 
meridian  for  the  hour  angle  from  noon  at  which  the  observa- 
tion is  taken,  the  sign  of  application  to  the  noon  constant 
being  the  same  as  that  of  the  altitude;  second,  a  correction 
representing  the  difference  of  latitude  for  the  run  for  the 
interval  of  time  in  the  hour  angle,  the  sign  of  which  is  +  if 
of  the  same  name  as  the  latitude  for  forenoon  observations, 
( — :)  for  afternoon  observations ;  ( — )  if  of  a  different  name 
from  the  latitude  for  forenoon  observations,  -(-  for  afternoon 
observations.  In  getting  this  latter  correction  it  is  well  to 
remember  that  if  for  a  given  speed  of  the  ship,  the  difference 
of  latitude  for  one  hour  is  x',  then  for  one  minute  it  is  x". 

In  preparing  the  "  constant/'  or  "  constants,"  to  be  used  for 
the  noon  latitude,  the  required  longitude  and  the  hour  angles 
for  the  reduction  of  observations  to  the  meridian  are  obtained 
in  the  following  way : 

Find  the  change  in  longitude  from  noon  of  the  preceding 
day  till  noon  of  the  given  day  by  using  the  run  till  11  a.  m. 

NOTE.— The  "  correction  for  difference  of  latitude,"  with  its  sign  as  determined 
above,  is  applied  algebraically  to  the  "  constant  for  latitude  by  meridian  altitude," 
the  sign  of  this  constant  being  regarded  as  +  when  the  sextant  altitude  is  subtracted 
from  it,  and  (— )  when  it  is  subtracted  from  the  altitude  to  give  the  latitude. 


CONSTANT  FOR  LATITUDE  533 

from  the  log  and  estimating  the  run  from  11  o'clock  to  noon; 
this  change  of  longitude,  expressed  in  time,  giving  the  amount 
that  the  deck  clock  must  be  set  ahead  or  back  in  order  that  it 
may  be  correct  at  meridian.  The  setting  of  the  clock  is 
usually  done  after  11  a.  m.,  and  this  shortening  or  lengthening 
of  the  last  hour  affects  the  ship's  run  for  that  hour. 

Then  find  the  longitude  at  local  apparent  noon  from  the 
a.  m.  longitude  and  the  run  in  longitude  during  the  interval, 
and  with  that  longitude  find  the  watch  time  of  noon  as  in 
Art.  198a.  Or,  having  learned  the  watch  error  on  local  ap- 
parent time  at  a.  m.  sight  (body  perhaps  on  or  near  the  prime 
vertical  at  that  time),  apply  to  it  the  difference  of  longitude 
in  time  from  sight  to  noon,  thus  getting  the  watch  error  at 
noon  on  local  apparent  time,  and  hence  the  watch  time  of 
noon.  From  the  watch  time  of  noon  and  the  watch  times  of 
observation,  the  hour  angles  are  found. 

Having  found  the  watch  time  of  noon,  reset  the  clock,  if 
necessary,  to  make  it  show  12  o'clock  when  the  watch  indi- 
cates local  apparent  noon. 

Ex.  193. — On  April  10,  1905,  a  vessel's  position  at  8  a.  m. 
(ship's  time)  was  Lat.  33°  57'  30"  K,  Long.  146°  38'  18"  B., 
and  by  sight  the  watch  was  slow  on  L.  A.  T.  6m  45s.  The  ship 
ran  till  apparent  noon  23°  (true)  12  knots  per  hour,  it 
being  estimated  that  the  clock  would  be  set  ahead  about  9 
minutes. 

It  is  required  to  find  the  longitude  at  noon  by  D.  E.,  the 
W.  T.  of  noon,  the  constant  for  latitude  by  meridian  altitude, 
given  the  I.  C.  +  1'  and  height  of  eye  45  feet. 

It  is  also  required  to  find  the  constants  which  will  give  the 
noon  latitude  when  applied  to  the  sextant  altitudes  observed 
at  the  following  intervals  of  time  before  noon,  20m,  15m,  10m, 
8m,  6m,  4m,  2m ;  also  the  W.  T.  of  the  first  observation  and  the 
noon  latitude,  if  at  the  first  observation  the  sextant  altitude 
of  the  sun's  lower  limb  was  62°  28'  50". 


534  NAUTICAL  ASTRONOMY 

D.  E.  8  a.  m.  to  noon. 

The  clock  being  set  ahead  9  minutes,  the  distance  run  will 
be  12  X  3.85  =  46.2  miles. 


True  Course. 
23° 

1       Distance.      1 
1             46.2             1 

Diff.  Lat.      1        Dep. 
42.5  N          1        18.1  E 

1     Diff.  Long. 
21'.9  E 

o    f  it 

o     t   n 

Lat.  8  a.  m. 

335780  N 

Long,  at  8  a.  m. 

146  38  18  E 

Diff.  of  Lat. 

4230  N 

Diff.  of  Long. 

2164  E 

Lat.  in  at  noon  by  D  R.  31  40  00  N  Long,  in  at  noon  by  D.  R.  147  00 12  E 

Lo  =    34°.3  N  =    9h.8  E 

(1)   To  find  the  W.  T.  of  noon. 

At  8  a.  m.  watch  slow  on  L.  A.  T.  6m  45s 

Change  in  time  due  to  Diff.  of  Long.  21'.9  E     1     27  .6 

At  noon  watch  slow  on  L.  A.  T.  8m  12S.6 

Therefore,  W.  T.  of  local  apparent  noon  is  llh  51m  47S.4  a.  m. 
and  W.  T.  of  first  observation  was  11  31    47  .4  a.  m, 


CONSTANT  FOR  LATITUDE 


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536  NAUTICAL  ASTRONOMY 

254.  Fourth  method.— By  altitude  of  the  pole  star.     The 

given   quantities   are   I,   d,   and  h,  the   required   one  is   L. 
Formulas  (206)  apply  here,  but  owing  to  a  very  small  polar 
distance  in  the  case  of  Polaris,  they  can  be  simplified, 
tan  </>  =  cot  d  cos  i, 
cos  <£'  =  cos  <j>  sin  h  cosec  df 
90°  —  L  =  <l>  ±  <#>'. 

As  before,  <J>  is  the  polar  distance  of  foot  of  perpendicular, 
<j>'  is  the  zenith  distance  of  foot  of  perpendicular. 
Now  <j>  and  p  are  so  small,  that  having  substituted  90° — p  for 
d,  90°  •  —  z  for  hf  we  may  consider  cos  <£  and  sec  p  each  unity, 
also  tan  <£  =  </>  tan  1'  and  tan  p  =  p  tan  1' ;  hence  the  above 
will  become 

$  —p  cos  t, 
cos  <£'=cos  z  or  <]>'=z, 
90°— £=<£±(/>'=</>±z= 

L—li—^  or  L=180° 
Therefore,  L  =  h  —  pcos  t,  (215) 

where  ft  is  the  true  altitude  of  Polaris; 

p,  the  polar  distance  of  the  star  at  the  instant  of  ob- 
servation ; 
it  its  hour  angle. 

Close  attention  must  be  given  to  the  sign  of  cos  i  as  it  affects 
the  sign  of  application  of  <£.  If  t  <  6  hours  or  >  18  hours, 
cos  t  is  -f ;  if  t  >  6  hours  and  <  18  hours,  cos  t  is  — ,  and 
L  =  h  +  p  cos  t. 

The  second  value  L  =  180°  —  0  —  h  is  inadmissible  as  it 
exceeds  90°. 

Since  by  definition  the  latitude  equals  the  altitude  of  the 
elevated  pole  equals  PN9  in  position  a,  L=N& — P&  =  h1 — p; 
at  position  b,  L  =  PN  —  N\>  +  bP  =  h2  +  p.  The  mean 
of  these  two  will  give  excellent  determinations,  that  is,  the 
mean  of  the  latitudes  from  observations  at  upper  and  lower 
transits.  (See  Fig.  117.) 


LATITUDE  BY  POLARIS 


537 


Let  M  be  any  position  of  Polaris  when  t  is  <  6  hours. 
Let  ZM  =  Zd.  Let  Mm  be  a  perpendicular  to  the  meridian, 
and  regard  PMm  as  a  plane  triangle,  then  </>  is  the  polar  dis- 
tance of  m  and  equals  p  cos  t.  By  the  above  formulae 
L  =  h  —  p  cos  t;  in  other  words,  Nm  is  assumed  equal  to 
Nd  or  HM,  the  star's  altitude.  For  any  other  position  as  M^ , 
when  t  is  >6  hours,  L  =  h  +  p  cos  t  and  -ZVra!  is  assumed 
equal  to  Nd^  or  H^M^ .  Though  these  assumptions  are  a 
source  of  a  slight  error,  the 
above  method  is  sufficiently  ex- 
act for  all  nautical  purposes. 
It  is  available  at  all  times  when 
the  horizon  is  distinctly  seen, 
and  the  star  Polaris  is  visible 
and  of  sufficient  altitude  to 
eliminate  the  errors  of  re- 
fraction. Its  application  is 
limited  to  the  northern  hemi- 
sphere. 

Table   IV  of   the   Nautical 
Almanac    gives    the    value   of 

p  cos  t  at  intervals  of  5  minutes  of  hour  angle,  computed  for 
a  mean  value  of  the  right  ascension  and  polar  distance  of 
Polaris  for  the  current  year.  This  correction,  applied  ac- 
cording to  its  sign,  will  give  the  latitude,  which  is  not  so 
accurate  as  that  computed  from  the  apparent  right  ascension 
and  polar  distance  from  the  Almanac,  except  when  these  are 
near  the  values  used  in  computing  the  table. 

In  Art.  176  of  Chauvenet's  Astronomy,  a  rigorous  formula 
is  deduced,  from  which  the  latitude  by  altitude  of  the  pole 
star  may  be  found  with  great  accuracy. 

It  is 

L  =  h—  pcoa  t  +  Jjo'sin  I" sin8 1  tan  h  | 

—  i  p*  sin2 1"  cos  t  sin2 1  +  i  p'  sin8 1"  sin4 1  tan8  h  ]  ( 


FIG.  117. 


538 


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NAUTICAL  ASTRONOMY 

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540  NAUTICAL  ASTRONOMY 

The  sum  of  the  last  three  terms  in  equation  216,  page  537, 
represents  dm  in  Fig.  117,  also  dm^ ,  etc. 

Table  28  of  Bowditch  is  computed  from  that  formula. 

The  last  two  terms  may  he  omitted  with  no  greater  error 
in  the  latitude  than  1".  If  p  cos  t  is  the  only  correction  ap- 
plied, the  error  will  amount  to  only  about  1'  when  t  =  6 
hours  and  li  =  54°,  and  a  maximum  of  3'  when  t  =  6  hours 
and  h  =  68°  30'. 

255.  Fifth  Method,  called  Chauvenet's  Method.  This  con- 
sists in  finding  the  latitude  by  two  altitudes  near  the  meridian 
when  the  time  is  not  known. 

It  frequently  happens  that  the  time  is  uncertain,  or  the 
deck  watch  has  not  been  compared  with  the  chronometers, 
enabling  the  navigator  to  get  the  correct  hour  angle  at  obser- 
vation; under  such  circumstances  this  method  is  of  great  use 
to  the  practical  navigator. 

Let  h^  and  Ji2  be  the  true  altitudes  of  the  body  at  the  first 

and  second  observations; 

TFj  and  W 2  be  the  corresponding  watch  times  of  ob- 
servation ; 

x  and  y  be  the  unknown  hour  angles  of  the  body,  re- 
spectively, at  the  first  and  second  observations; 
T  be  the  interval  of  time  between  the  observations, 

then!7  =  W2  —  F±; 

x  ~  y  be  the  difference  of  hour  angles  of  the  body  at 
the  two  observations.  For  the  sun,  it  is  an  inter- 
val of  apparent  time  and  without  error  may  be 
represented  by  T.  For  a  star  x  ~  y  is  an  inter- 
val of  sidereal  time  which  equals  T  when  T  is 
reduced  by  Table  III  to  a  sidereal  interval. 


CHAUVENET'S  METHOD  541 

As  in  Art.  251,  let  7i0  represent  the  true  meridian  altitude 
of  the  body,  and  A0/t  the  change  in  altitude  in  lm  from  the 
meridian. 

Then,  by  formula  (213), 


Taking  the  half  sum  of  the  above  equations,  we  have 

»»  =  *!_+_*»  +  *+jCAA.     '        (817) 

but 

£+jf  =  (^y\*4.(*—y\*-_(*+y\*i(T\*. 

*     (  ».  /jv>^~\^/-  \T/ 

therefore, 


Taking  the  difference  of  the  same  equations,  we  have 

(x  —  y)  2  AO£ 


and 

7t-A    ._ 

(219) 


/ 
Substituting  this  value  of  — £  ^in  (218), 


Therefore,  to  obtain  the  meridian  altitude  by  this  method, 
two  corrections  must  be  added  to  the  mean  of  the  body's  two 
true  altitudes.  The  first  is  of  the  form  of  the  reduction  to 


542  NAUTICAL  ASTRONOMY 

the  meridian,  using  one-half  the  elapsed  time  in  place  of  the 
hour  angle.  The  second  is  the  square  of  one-fourth  the  dif- 
ference of  the  altitudes  divided  by  the  first  correction,  care 
being  taken  to  have  both  terms  of  this  fraction  in  the  same 
unit,  usually  seconds  of  arc. 

The  second  correction  is  the  larger  of  the  two,  as  a  gen- 
eral thing,  and,  as  this  depends  largely  on  the  difference  of 
altitudes,  the  accuracy  of  the  resulting  latitude  will  depend  on 
the  precision  with  which  the  altitudes  have  been  measured, 
since  errors  due  to  the  tabulated  dip,  refraction  and  constant 
instrumental  errors  affect  both  altitudes  alike. 

Having  found  TiQ>  proceed  as  in  Art.  251  to  find  the  latitude. 

When  Ji2  =  Tilf  the  second  correction  reduces  to  zero; 
therefore,  the  most  favorable  case  is  that  of  equal  altitudes 
observed  on  each  side  of  the  meridian. 

The  value  of  the  hour  angles  may  be  obtained  approxi- 
mately thus, 

From  (218),  X         —  *  &*  ~  &i) 


_^(h,  —  hl).    T 

|A0£  2 


f 


Restrictions. — The  restrictions  of  this  method  are  the  same 
as  those  limiting  the  reduction  of  a  single  altitude  to  the 
meridian.  It  must  be  remembered,  however,  that  the  obser- 
vations in  this  method  are  not  made  at  the  same  place.  A 


CHAUVENET'S  METHOD  543 

slight  change  of  the  observer's  zenith,  which  would  result 
from  a  small  interval  between  observations,  would  produce  but 
a  slight  error  and  especially  so  when  the  course  is  at  a  right 
angle  to  the  bearing  of  the  body.  When  the  interval  is  com- 
paratively large,  and  the  distance  run  also  of  consequence, 
the  first  altitude  must  be  reduced  for  the  run  (see  Art.  213) 
to  what  it  would  have  been,  if  observed  at  the  same  time  at 
the  second  position.  The  value  of  T  will  not  change. 

The  latitude  found  will  be  that  at  the  instant  of  the  second 
observation;  and  to  obtain  the  noon  latitude,  allowance  must 
be  made  for  the  change  in  latitude  during  the  run  from  the 
time  of  the  second  observation  to  noon. 

It  is  not  necessary  to  reduce  each  altitude  to  a  true  altitude 
and  then  take  the  mean.  It  will  be  sufficiently  accurate  for 
practical  purposes  at  sea  to  take  the  mean  of  the  sextant  alti- 
tudes, and  reduce  it  to  the  true  altitude  of  the  center. 


514 


NAUTICAL  ASTRONOMY 


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PRESTEI/S  METHOD  545 

256.  Sixth  method.-— To  find  the  latitude  by  the  rate  of 
change  of  altitude  near  the  prime  vertical  (called  Prestel's 
method).  In  Art.  237,  by  differentiation  of  the  fundamental 
formula  of  the  astronomical  triangle, 

sin  h  =  sin  L  sin  d  -)-  cos  L  cos  d  cos  t, 

regarding  h  and  t  as  variables,  we  found  formula  (188),  from 
which,  expressing  dt  in  time,  we  have 

dt  = jg-  sec  L  cosec  Z, 

in  which  dh  is  a  small  change  of  altitude  in  seconds  of  arc, 
occurring  in  a  very  brief  interval  in  seconds  of  time. 

If  the  altitude  is  increasing,  as  when  the  body  is  East  of 
the  meridian,  the  hour  angle  is  diminishing  or  dt  is  ( — ) ;  if 
the  altitude  is  diminishing,  as  when  the  body  is  West  of  the 
meridian,  the  reverse  holds  true. 

Let  wi  be  the  noted  time  when  the  body  is  at  the  alti- 
tude &! , 
w2  that  when  the  body's  altitude  is  Ji2'} 

then        T  =  — (w±  —  w2)  = *-= — -1  sec  L  cosec  Z, 


and          T  =  w2  —  w±  *  .. ,   'l  sec  L  cosec  Z. 

cos  L  =  ^^  cosec  Z.  (223) 

When  Z  is  near  90°,  its  cosecant  varies  very  slowly  and  when 
Z  =  90°  we  have 

cos  L  =  h^pl  *  (224) 

The  accuracy  of  this  method  depends  on  the  precision  with 

which  the  altitudes  are  measured  and  the  care  with  which 

times  are  noted. 

As  the  latitude  is  found  from  its  cosine,  the  method  is 

more  precise  in  high  than  in  low  latitudes.     Though  the  re- 


546  NAUTICAL  ASTRONOMY 

suit  may  be  only  approximate,  it  may  be  useful  in  restricting 
the  ship's  position  to  a  limited  portion  of  a  Sumner  line. 

The  time  when  a  body  is  on  the  prime  vertical  can  be  found 
from  the  azimuth  tables,  or  from  Art.  239,  or  sufficiently  near 
by  compass  if  its  error  is  known,  or  by  Table  C  of  N".  A. 

In  case  the  body  is  within  2°  of  the  P.  V.,  measure  the  alti- 
tudes and  note  the  times  carefully,  not  letting  T  be  >  8m; 
use  formula  (224:)  and  for  high  latitudes  the  result  may 
be  found  within  a  limit  of  error  that  would  still  make 
it  desirable.  However^  only  an  emergency  will  justify  the 
use  of  this  method.  Chauvenet  recommends  bringing  one 
reflected  limb  of  the  sun  into  contact  with  the  sea  horizon, 
the  time  being  noted ;  then,  keeping  the  sextant  clamped,  note 
the  time  when  contact  of  the  other  limb  occurs ;  beginning  in 
the  forenoon  with  the  upper  limb,  in  the  afternoon  with  the 
lower  limb;  dh  will  be  the  sun's  diameter  in  seconds  from 
the  Almanac. 

In  case  the  body  is  more  than  2°  from  the  P.  V.,  use  for- 
mula (223). 

Ex.  198.— April  24,  1905,  in' approximate  latitude  43°  20' 
N.9  longitude  30°  10'  W.,  about  5  p.  m.,  the  sun  bearing  true 
West,  the  sun's  reflected  lower  limb  was  brought  tangent  to 
the  horizon.  W.  T.  4h  59m  03s.  The  sextant  being  kept 
clamped,  when  the  upper  limb  made  contact  with  the  horizon, 
the  watch  read  5h  Olm  58s.  Find  the  latitude. 


Formula 

i 

**~*i           h  ^ 

15 
T=    175« 
L  -  43°  16 

3                 log       3.28133 
colog  8.82391 
colog  7.75696 

15  T 
fcj^  Aj  =  sun's  diameter 

=  (15'  55".66)  X  2 
=  1911".32 

r  15"  N     cos      9.86220 

257.  Reduction  of  latitude. — In  the  previous  articles  of 
this  chapter,  we  have  assumed  the  earth  to  be  a  sphere,  im- 
plying that  a  plumb-line  at  any  point  of  the  earth's  surface, 


"KEDUCTION  OF  THE  LATITUDE" 


547 


if  extended,  passes  through  the  earth's  center,  and  that  the 
altitude  of  an  observed  body,  after  the  usual  corrections  have 
been  applied,  is  referred  to  the  center. 

The  earth  is  not  a  sphere  but  a  spheroid,  and  the  vertical 
line  at  any  point  of  the  surface  as  O'L  in  Fig.  118,  which 
corresponds  exactly  with  the  normal  drawn  at  that  point,  doe? 
not  coincide  with  the  earth's 
radius    passing    through    the 
same   point   excepting   at  the 
equator  and  at  the  poles. 

The  point  Z  where  the  ver- 
tical line  O'L  prolonged  meets 
the  celestial  sphere  is  the  geo- 
graphical zenith  and  the  angle 
ZO'Q  is  the  geographical  lati- 
tude of  the  point  L,  as  deter- 
mined by  observations  at  sea. 
The  point  Z'  in  which  the 

radius  OL  prolonged  meets  the  celestial  sphere,  is  the  geocen- 
tric zenith  and  the  angle  Z'OQ  is  the  reduced  or  geocentric 
latitude. 

The  geocentric  latitude  is  smaller  than  the  geographical 
latitude  at  all  places  except  at  the  equator  and  poles  where 
they  are  equal ;  the  difference  between  the  two  being  the  angle 
OLO'  called  the  "  angle  of  the  vertical "  or  the  "  reduction  of 
the  latitude." 

Though  necessary  in  certain  refined  observations  ashore,  it 
is  not  necessary  to  consider  the  reduction  at  sea  where  ex- 
treme- precision  is  unattainable. 


FIG.  118. 


CHAPTEE  XVIII. 

CHRONOMETER  ERROR,  CORRECTION,  AND  RATE.— 
LONGITUDE  ASHORE  AND  AT  SEA. 

258.  It  has  already  been  shown  in  Chap.  X  that  the  chro- 
nometer is  the  navigator's  means  of  getting  the  Greenwich 
mean  time  of  any  desired  instant  or  observation.     Though 
constructed  with  the  greatest  care  and  at  much  expense  it  is 
far  from  perfect,  seldom  indicating  the  exact  time  of  the 
prime  meridian  and  seldom  running  with  regularity  for  any 
length  of  time. 

However,  a  sidereal  or  a  mean  time  chronometer  is  said  to 
be  regulated  to  local  or  Greenwich  time,  when  its  error  on  that 
particular  time,  the  amount  by  which  it  is  fast  or  slow  of  that 
time,  and  its  rate,  or  daily  gain  or  loss,  are  known. 

Both  the  error  and  rate  are  positive,  if  the  chronometer  is 
fast  and  gaining;  otherwise,  negative;  the  sign  of  the  error 
being  the  sign  of  application  to  the  correct  standard  of  time 
to  get  the  chronometer  reading.  It  is  preferable,  however, 
to  regard  the  error  as  a  correction  to  be  applied  to  the  chro- 
nometer reading  to  obtain  the  desired  true  time,  and  to  con- 
sider the  rate  as  a  daily  change.  Both  are  positive  or  plus 
when  the  chronometer  is  slow  and  losing. 

259.  To  find  the  rate.— The  rate  is  found  by  taking  the 
algebraic  difference  (that  is,  the  numerical  difference  when 
of  the  same  name,  the  numerical  sum  when  of  a  different 
name)   of  the  errors  on  two  different  days  and  dividing  it 
by  the  elapsed  time  in  days  and  decimals  of  a  day.     The 
interval  should  be  at  least  5  to  7  days.    When  the  errors  are 


SEA  KATE  549 

determined  at  two  different  places,  the  times  of  observation 
should  be  reduced  to  one  (say  Greenwich)  meridian  and  the 
interval  found  from  the  two  reduced  times. 

The  rate  will  be  gaining  when  both  errors  are  fast  and  the 
last  one  is  the  greater,  when  both  errors  are  slow  and  the  last 
one  the  lesser  of  the  two,  or  when  the  error  changes  from 
a  slow  to  a  fast  one;  otherwise,  the  rate  will  be  a  losing  one. 
Ex.  199. — The  error  of  a  given  chronometer  on  G.  M.  T.  on 
April  15  at  noon  was  -|-  5m  32S.5;  at  noon  on  April  25  it  was 
+  5m  35S.8.  Eequired  the  daily  rate. 

Error  at  noon  April  15,  +  5m  32S.5 
Error  at  noon  April  25,  +  5  35  .8 
Change  for  10  days  +  33.3 
Daily  rate  +  0 .33 

260.  To  find  the  error  on  a  given  date,  knowing  the  error 
on  another  date  and  the  daily  rate. — Multiply  the  daily  rate 
by  the  number  of  days  elapsed  since  the  determination  of  the 
error  and  this,  applied  with  proper  sign  to  the  original  error, 
will  give  the  error  on  the  required  date. 

Ex.  200. — With  the  data  of  the  above  example,  find  the  error 
of  the  same  chronometer  on  G.  M.  T.  at  noon  April  30. 
Error  April  25,  +  5m  35S.8       Daily  rate    +  Os.33 
Change  +          1  .65      No.  of  days  5 

Error  April  30,  +  5m  37S.45      Change         +  ls.65 

261.  Sea  rate. — Ordinarily  the  error  and  rate  of  a  chro- 
nometer are  determined  entirely  from  shore  determinations, 
and  that  error  is  brought  up  by  its  rate  to  the  instant  of  later 
observations  at  sea  in  working  for  longitude.     Now  this  rate 
found  in  port  may  be  very  different  from  the  actual  sea  rate, 
even  at  the  same  temperature. 

In  case  an  error  is  determined  just  before  leaving  port  and 
again  after  return  to  the  same  port,  the  difference  of  errors 
divided  by  the  elapsed  time  will  give  the  sea  rate. 


550  NAUTICAL  ASTRONOMY 

Again,  a  vessel  on  a  voyage  may  stop  at  many  places  whose 
longitudes  are  well  known,  these  having  been  determined  per- 
haps by  direct  or  indirect  telegraphic  connection  with  Green- 
wich or  some  place  of  known  longitude. 

Say  the  error  of  the  chronometer  on  G.  M.  T.  is  found  at 
place  A,  of  known  longitude;  by  applying  the  longitude  to 
the  local  time  of  determination  of  the  error,  the  Greenwich 
time  of  the  determination  is  gotten.  At  place  E,  obtain  the 
same  data,  the  chronometer  error  on  G.  M.  T.,  and  the  Green- 
wich instant  corresponding.  The  algebraic  difference  of  the 
errors,  divided  by  the  elapsed  Greenwich  time,  will  be  the  sea 
rate,  the  rate  being  regarded  as  uniform. 

Ex.  201. — On  April  2,  1905,  at  Southampton,  in  longitude 
1°  18'  W.,  a  time  ball  was  dropped  at  Oh  00m  00s  L.  M,  T.  At 
this  instant  the  error  of  a  chronometer  A  was  found  to  be 
slow  14m  52s  on  G.  M.  T. 

On  April  28,  1905,  at  Lisbon,  in  longitude  9°  12'  W.  by 
single  altitudes  in  the  forenoon,  using  artificial  horizon,  the 
error  of  the  same  chronometer  was  found  to  be  slow  of  G.  M.  T. 
8m  50S.005.  C.  T.  of  observation  9h  15m  36S.2.  Eequired  the 
sea  rate. 

hms  hms  ms 

April  2,  L.M.T.  0  00  00    April  28,  C.  T.        9  15  36.2    Error  April  2,  slow  14    52 
Long.    W  613    C.C.  +850       Error  April  28,  slow  8    50.005 

G.M.T.  April  2,  0  05  12    April  27,  G.M.T.  21  24  26.2    Gain  =    6    01.995 

April  2,  G.M.T.      0  05  12  =      361».996 

Elapsed  time  25<i  21'1 19'"  14-.2 

=    25d.888 

Rate  or  Daily  Gain  =  3§^?    =    13«.988. 


262.  Irregular  rate. — In  case  the  rate  is  not  constant,  as 
shown  by  getting  two  different  rates  by  observations  for  two 
different  intervals,  the  change  in  the  rate  itself  may  be  re- 
garded as  uniform  and  the  rate  interpolated  to  the  middle  in- 
stant between  the  two  intervals.  This  will  permit  the  use  of 


SEA  RATE  551 

the  formula  for  uniformly  accelerated  or  retarded  motion  in 
finding  the  change  in  error  for  a  given  interval  of  time. 
The  first  rate  is  taken  as  correct  at  the  middle  instant  of  the 
first  interval. 

At  the  second  series  of  observations,,  the  second  rate  is 
taken  as  at  the  middle  instant  of  the  second  interval. 

Ex.  202. — The  error  of  a  chronometer  B  at  noon  April  1, 
1905,  at  St.  Nicholas  Mole,  Hayti,  was  found  to  be  fast  of 
G.  M.  T.  llm  42S.5;  at  same  place,  at  noon  April  11  fast  of 
G.  M.  T.  llm  51s. 

The  chronometer  was  then  carried  to  a  point  on  the  South 
coast  of  Cuba  where  observations  for  longitude  were  made  in 
the  forenoon  and  afternoon  of  April  16. 

On  return  to  the  Mole,  the  error  at  noon  April  22  of  the 
same  chronometer  was  found  to  be  fast  of  G.  M.  T.  llm  56S.4, 
and,  again,  on  April  30  fast  of  G.  M.  T.  llm  588.4.  Required 
the  error  at  noon  April  16. 

m    s  m     s 

Error  April  11,         +    11  51  Error  April  30,          11  58.4 

Error  April  1,  +    1142.5  Error  April  22,         1156.4 


Change  in  10  days  +  8.5  Change  in  8  days  +          2 

Daily  rate  +  0.85         Daily  rate  +         0.25 

These  rates  are  assumed  correct  at  the  middle  instant  of 
the  period  during  which  they  were  determined. 
April  6,  rate  +  Os.85 

April  26,  rate  +  0  .25 

Change  of  rate  in  20  days  —  Os.60 
Daily  change  of  rate  —  0  .03 

Rate  April  11,  +0.70 

The  problem  now  resolves  itself  into  this,  "  On  April  11  the 
error  of  a  chronometer  was  -f-  llm  51s,  the  daily  rate  -f-  0s. 70, 
retardation  of  the  rate  Os.03.     Find  the  error  April  16." 
Taking  the  formulae 

8  =  (70  +  4  oO*  and  E  =  E0  +  S,  (225) 


552  NAUTICAL  ASTRONOMY 

where  8  is  the  change  in  error  in  time  t, 

70  is  the  initial  rate  April  11  +  Os.70, 
a  the  daily  retardation  equals  Os.03, 
t  the  elapsed  time  equals  5  days, 
E0  the  initial  error  April  11  equals  +  llm  51s, 
E  the  required  error  April  16. 


S  =  +    08.70  _'  5==+  (o«  70  -  0-.075)  5  = 

E  =  llm  51s  +  3S.125  —  +  llm  54M25;  or  at  noon  April  16, 
the  chronometer  is  fast  of  G.  M.  T.  llm  54M25. 

As  the  error  on  April  16  is  to  be  used  in  the  determination 
of  longitude,  it  should  also  be  determined  by  working  back 
from  April  22,  and  the  two  values  combined  by  giving  weights 
to  each  inversely  proportional  to  its  interval  of  time  from  the 
original  determination. 

From  the  rate  on  April  26  find  the  rate  on  April  22,  call  it 
VQ  ,  and  let  E0  be  the  error  April  22  ;  then,  by  substitution  as 
before,  we  shall  obtain  an  error  of  the  chronometer,  on  April 
16,  of  llm  53S.64. 

There  is  a  discrepancy  in  the  two  errors  arising  from  the 
fact  that  the  actual  change  of  rate  is  not  uniform  as  assumed. 

If  E!  is  an  error  brought  forward  tf±  days  ; 
E2  ,  an  error  carried  back  t2  days  ; 
Ex  ,  the  probable  error  at  the  given  time  ; 
then,  by  the  method  of  "  Least  Squares," 


Substituting  in  the  above  equation 

E!  =  +  llm  54M25,  t±  —  5,  E2  =  +  llm  53S.64,  tt  =  6, 
we  have 

5(_0S.485) 
E,  =  +  llm  54M25  +     v         -  -  =  +  llm  538.905. 

Therefore  the  chronometer  is  fast  of  G.  M.  T.  llm  53S.905  on 


CHRONOMETER  ERROR  553 

April  16;  that  is  to  say  the  probable  C.  C.  is  (— )  llm  538.905 
on  that  date. 

However,  as  chronometer  rates  are  greatly  affected  by  varia- 
tions in  temperature,  the  theory  of  this  method,  that  of  a 
uniform  change  of  rate,  is  not  tenable  when  there  have  been 
erratic  changes  of  temperature  between  successive  ratings. 

263.  Finding  the  chronometer  error. — Before  leaving  port, 
the  navigator  must  ascertain  the  error  and  rate  of  each  of  his 
chronometers,  and,  if  he  has  sufficient  data,  construct  a  tem- 
perature curve. 

The  methods  of  obtaining  chronometer  error  and  rate  may 
be  considered  under  three  general  heads : 

( 1 )  Observatory  methods,  as  by  transits. 

(2)  By  time  signals. 

(3)  By  the  navigator's  own  observations. 

264.  (1)  Observatory  methods. — The  most  accurate  method 
of  finding  chronometer  error  is  by  noting  the  chronometer 
time  of  transit  of  a  heavenly  body  across  each  wire  of  a  transit 
instrument  well  adjusted  in  the  meridian.    The  mean  of  these 
times,  reduced  to  the  time  of  meridian  passage  by  applying 
the  proper  corrections  in  case  the  middle  wire  is  not  exactly 
in  the  meridian,  will  give  the  chronometer  time  of  transit. 

At  the  instant  of  transit  of  a  star  over  the  upper  branch 
of  the  meridian,  its  right  ascension  equals  the  local  sidereal 
time;  if  over  the  lower  branch  of  the  meridian,  the  L.  S.  T. 
equals  the  star's  right  ascension  plus  12  hours.  If  the  chro- 
nometer is  a  sidereal  chronometer,  the  difference  between  its 
reading  at  the  instant  of  the  star's  transit  and  the  star's  right 
ascension  will  be  the  error  on  L.  S.  T.  If  the  star  is  observed 
at  the  lower  transit,  then  the  error  on  L.  S.  T.  will  be  the 
difference  between  the  chronometer  reading  and  the  sum  of 
the  star's  right  ascension  plus  12  hours. 

If  the  chronometer  is  a  mean  time  chronometer,  convert  the 
local  sidereal  time  of  transit  into  local  mean  time,  apply  the 


554  NAUTICAL  ASTRONOMY 

longitude  to  obtain  the  G.  M.  T. ;  the  difference  hetween  which 
and  the  chronometer  time  of  the  transit  will  be  the  error  of 
the  chronometer  on  G.  M.  T. 

The  time  of  upper  transit  of  the  sun's  center  is  Oh  00m  00s 
of  apparent  time;  the  corresponding  local  sidereal  time  must 
be  found,  if  the  time  of  transit  was  marked  by  a  sidereal  chro- 
nometer to  find  its  error  on  L.  S.  T. ;  or,  the  corresponding 
G.  M.  T.  must  be  found,  if  the  time  of  transit  was  marked  by 
a  mean  time  chronometer  whose  error  on  G.  M.  T.  is  desired. 

Transits  of  stars  are  preferred  to  transits  of  the  sun.  In 
this,  as  in  all  other  cases  of  finding  time,  observers  are  ad- 
vised not  to  use  the  moon. 

By  repeating  the  observations,  at  a  subsequent  time,  the 
rate  will  be  found  as  in  Art.  259.  Though  the  error  may  not 
be  good  when  the  instrument  is  somewhat  out  of  adjustment, 
the  rate  will  be  good,  if  the  second  error  is  determined  by  the 
same  instrument,  without  change  of  position  or  adjustment, 
and  under  like  circumstances. 

265.  (2)  By  time  signals. — At  nearly  all  important  sea  ports 
of  the  world  a  time  signal  is  made  at  a  specified  instant  of 
time.  This  instant  will  be  found  in  the  sailing  directions 
of  the  locality  and  in  the  daily  papers ;  the  latter  usually  pub- 
lish the  day  following  any  failure  or  error  in  the  signal. 
The  general  form  of  signal  is  a  time  ball  or  gun-fire.  The 
ball  is  usually  painted  black  and,  a  few  minutes  before  the 
instant  of  dropping,  is  hoisted  to  the  top  of  a  high  pole  con- 
spicuously located  so  as  to  be  seen  from  all  parts  of  the  har- 
bor, and  is  dropped  by  electricity,  usually  by  signal  from  an 
observatory.  The  observer  notes  the  time  by  his  chronom- 
eter the  moment  the  ball  starts  from  the  top;  the  difference 
between  the  chronometer  face  and  the  L.  M.  T  of  fall  is  the 
error  on  L.  M.  T.,  from  which  the  error  on  G.  M.  T  is  at  once 
found  by  applying  the  longitude. 

In  cases  where  the  signal  is  gun-fire,  the  gun  is  usually 


CHRONOMETER  ERROR  555 

fired  electrically  from  an  observatory.  The  flash  is  the  mo- 
ment to  be  timed  by  the  chronometer;  but  if  not  seen,  listen 
for  the  report,  making  an  allowance  for  the  velocity  of  sound 
at  the  rate  of  1090  feet  per  second  at  a  temperature  of  32°  F., 
with  an  increase  or  decrease  for  each  degree  above  or  below 
32°  F.  at  the  rate  of  1.15  of  a  foot  per  degree.  Dividing  the 
known  distance  between  the  observer  and  the  gun  by  the 
proper  velocity  of  sound  per  second,  we  find  the  correction  in 
seconds  to  be  subtracted  from  the  chronometer  reading  at 
hearing  the  signal  to  give  the  chronometer  face  at  the  flash 
or  time  of  signal. 

Ex.  208. — At  Southampton,  England,  a  time  ball  was 
dropped  on  April  10,  1905,  at  lh  00m  00s  G.  M.  T.,  a  chro- 
nometer at  the  instant  reading  llh  45m  30s.  What  was  the 
chronometer  error  ? 

G.  M.  T.  lh  00ra  00s 

Chronometer  face       11    45     30 


Chronometer  error  —  lh  14m  30s 

In  other  words,  the  chronometer  is  slow  on  G.  M.  T.  lh  14m 
30s;  or,  the  C.  C.  is  +  lh  14m  30s. 

Standard  time  (Art.  177)  is  now  being  used  in  many  coun- 
tries ;  England  keeps  Greenwich  time  and  France  Paris  time ; 
Germany,  Austria,  Italy,  and  Denmark  have  adopted  the 
standard  time  of  the  meridian  of  15°  E.  The  United  States 
have  adopted  the  standard  times  as  explained  in  Art.  177; 
Cape  Colony  the  time  of  the  meridian  of  22°  30'  E. 

Clocks  in  business  houses,  hotels,  and  schools,  when  elec- 
trically controlled,  are  thrown  into  circuit  with  the  local  tele- 
graph lines,  and  are  corrected  electrically  at  noon. 

Navigators,  being  at  the  telegraph  office  when  the  time 
signals  are  being  made,  can  find  with  ease  and  certainty  the 
error  of  a  chronometer  on  the  standard  time  being  received, 
by  noting  the  chronometer  face  when  the  signal  is  made. 


556  NAUTICAL  ASTRONOMY 

The  difference  between  the  chronometer  face  and  the  time 
of  the  signal  will  be  the  error  on  that  time.  The  error  on 
G-.  M.  T.  may  be  obtained  by  applying  to  this  error  the  proper 
difference  of  longitude. 

For  a  list  of  ports  in  which  time  signals  are  made  and  a 
description  of  said  signals,  see  "  Lecky's  Wrinkles,"  p.  567, 
13th  edition. 

The  United  States  System  of  Time  Signals. 

In  the  United  States  time  signals  are  sent  not  only  over 
telegraph  wires  to  the  various  towns  and  cities  of  the  country, 
but  from  the  Government  wireless  stations  along  the  coast  to 
vessels  at  sea.  The  general  scheme  of  transmission  is  ex- 
plained in  the  following  extract  from  the  Annual  Eeport  of 
the  Superintendent  U.  S.  Naval  Observatory  for  the  fiscal 
year  ending  June  30,  1902 : 

Telegraphic  Time  Signals. 

Sent  out  at  noon  daily,  except  Sundays  and  holidays,  by  the 
U.  S.  Naval  Observatory. 

The  entire  series  of  noon  signals  sent  out  daily  over  the 
wires  is  shown  graphically  in  the  accompanying  diagram. 
This  represents  the  signals  as  they  would  be  recorded  on  a 
chronograph,  where  a  pen  draws  a  line  upon  a  sheet  of  paper 
moving  along  at  a  uniform  rate  beneath  it  and  is  actuated  by 
an  electro-magnet  so  as  to  make  a  jog  at  every  tick  of  the 
transmitting  clock.  The  electric  connections  of  the  clock  are 
such  as  to  omit  certain  seconds,  as  shown  by  the  breaks  in  the 
record.  These  breaks  enable  anyone  who  is  listening  to  a 
sounder  in  a  telegraph  or  telephone  office  to  recognize  the 
middle  and  end  of  each  minute,  especially  the  end  of  the  last 
minute,  when  there  is  a  longer  interval  that  is  followed  by 
the  noon  signal.  During  this  last  long  interval,  or  10-second 


U.  S.  TIME  SIGNALS 


557 


break,  those  who  are  in  charge 
of  time  balls  and  of  clocks  that 
are  corrected  electrically  at 
noon  throw  their  local  lines  in- 
to circuit  so  that  the  noon  sig- 
nal drops  the  time  balls  and 
corrects  the  clocks. 

This  series  of  noon  signals  is 
sent  continuously  over  the 
wires  all  over  the  United  States 
for  an  interval  of  five  minutes 
immediately  preceding  noon. 
The  transmitting  clock  that 
sends  out  the  signals  is  cor- 
rected very  accurately,  shortly 
before  noon,  from  the  mean  of 
three  standard  clocks  that  are 
rated  by  star  sights  with  a 
meridian  transit  instrument. 
The  noon  signal  is  seldom  in 
error  to  an  amount  greater 
than  one  or  two-tenths  of  a 
second,  although  a  tenth  more 
may  be  added  by  the  relays  in 
use  on  long  telegraph  lines! 
Electric  transmission  over  a 
continuous  wire  is  practically 
instantaneous.  For  time  sig- 
nals at  other  times  than  noon, 
similar  signals  can  be  sent  out 
by  telegraph  or  telephone  from 
the  same  clock  that  sends  out 
the  noon  signal. 


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ERROR  BY  SINGLE  ALTITUDES  559 

266.  (3)  By  the  navigator  using  his  own  observations. — 

The  navigator,  thrown  on  his  own  resources,  may  rate  his 
chronometers  by  one  of  the  following  methods,  using  the 
navigational  instruments  ordinarily  provided  him:  (a)  By 
single  altitudes,  (b)  Double  altitudes,  (c)  Equal  altitudes. 
In  all  the  methods  that  follow,  the  longitude  of  the  observa- 
tion spot  must  be  accurately  known  to  get  the  chronometer 
error  on  G.  M.  T.  The  latitude  of  the  spot  should  also  be 
accurately  known,  though,  if  the  body  is  observed  on  the  prime 
.vertical,  the  errors  due  to  uncertainties  of  latitude  will  be  a 
minimum.  In  single  or  double  altitudes,  the  closest  possible 
approximation  should  be  made  to  the  chronometer  error,  and 
this  used  in  finding  the  G.  M.  T.  for  which  to  take  out  the 
body's  declination.  In  the  absence  of  an  artificial  horizon,  a 
fair  error  may  be  found  from  a  good  sea  horizon  by  single  or 
double  altitudes.  Altitudes  of  any  heavenly  body  may  be 
used,  but  only  those  of  the  sun,  and  of  certain  stars,  are  recom- 
mended. 

(a)  Single  Altitudes. 

267.  With  the  sextant  and  artificial  horizon  at  a  place  of 
known  latitude  and  longitude,  the  navigator  takes  a  series  of 
altitudes  of  the  sun  or  a  star,  when  the  body  is  near  or  on 
the  prime  vertical,  noting  the  time  of  each  observation.     It  is 
recommended  to  take  a  series  of  five  altitudes  at  regular  in- 
tervals of  10'  of  sextant  arc ;  making  the  limbs  in  case  of  the 
sun  overlap,  and  noting,  by  chronometer  or  a  watch  com- 
pared with  a  chronometer  of  known  error,  the  instant  of  sepa- 
ration of  the  lower  limb  of  the  sun  and  the  upper  limb  of  its 
image  in  the  horizon  during  a.  m.  observations;  or,  by  a  re- 
verse operation,  noting  the  instant  of  contact  during  p.  m. 
observations.     The  mean  of  the  altitudes  and  the  mean  of 
the  times  should  be  taken. 

Using  an  approximate  chronometer  correction,  an  approxi- 
mate G.  M.  T.  of  observation  is  obtained,  the  declination  of 


560  NAUTICAL  ASTKONOMY 

the  body  is  taken  out,  and  the  altitude  reduced  to  a  true  alti- 
tude of  the  center. 

The  navigator  then  has  given  Id,  d,  and  L  from  which  to 
find  the  body's  hour  angle  as  in  Art.  226.  In  case  of  a  star, 
the  local  sidereal  time  obtained  from  the  star's  hour  angle  is 
reduced  to  L.  M.  T.,  to  which  the  known  longitude  is  applied ; 
the  difference  between  the  resulting  G.  M.  T.  and  the  chro- 
nometer reading  (or  chronometer  face)  at  the  instant  of  ob- 
servation will  be  the  error  of  the  chronometer  on  G.  M.  T. 

In  case  the  body  is  the  sun,  its  hour  angle,  reckoned  posi- 
tively, is  the  L.  A.  T.,  and  the  following  procedure  is  recom- 
mended: Apply  the  known  longitude  to  the  L.  A.  T.,  find- 
ing the  G.  A.  T.,  for  which  take  out  from  page  I,  N".  A.,  the 
equation  of  time.  Apply  the  equation  of  time  with  its  proper 
sign  to  the  G.  A.  T. ;  the  result  will  be  the  G.  M.  T.,  the  dif- 
ference between  which  and  the  chronometer  reading  at  the 
instant  of  observation  (C.  F.)  will  be  the  error  of  the  chro- 
nometer on  G.  M.  T.,  fast  or  slow,  according  as  the  chronom- 
eter face  is  greater  or  less  than  the  G.  M.  T. 

If  an  artificial  horizon  is  used,  as  it  should  be  when  pos- 
sible, two  sets  of  observations  should  be  made  with  a  different 
end  of  the  roof  in  each  set,  and  a  mean  of  the  two  resulting 
errors  taken  as  the  correct  error,  thus  eliminating  errors  due 
to  a  possible  want  of  parallelism  of  the  faces  of  the  glass. 

Such  observations  and  the  results  are  liable  to  the  same 
errors  as  are  similar  observations  for  longitude  at  sea. 

To  determine  the  rate. — On  a  subsequent  date,  repeat  the 
observations,  find  a  second  error,  take  the  algebraic  difference 
between  this  and  the  first  error,  and  divide  by  the  interval  in 
days  to  obtain  the  rate. 

The  observations  should,  as  far  as  possible,  be  taken  under 


DETERMINING  THE  KATE  561 

similar  conditions  as  to  body,  hour  angle,  altitude,  instru- 
ment, and  atmosphere.  If  like  conditions  exist,  each  error 
will  be  similarly  affected,  and  the  rate  should  be  reliable. 

Hence,  it  may  be  laid  down  as  a  general  rule  that  the  rate 
should  be  determined  by  a  comparison  of  a.  m.  sights  with 
a.  m.  sights,  or  p.  m.  sights  with  p.  m.  sights,  and  never  the 
reverse,  unless  conditions  absolutely  demand  it.  Under  such 
circumstances,  the  same  observer,  using  the  same  instrument, 
will  find  that  assumed  errors  of  latitude,  constant  instru- 
mental and  personal  errors  will  but  slightly  affect  the  rate. 


562 


NAUTICAL  ASTRONOMY 


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EQUAL  ALTITUDES  563 

(b)  By  Double  Altitudes  or  Altitudes  on  Opposite  Sides  of 
the  Meridian. 

268.  Instead  of  relying  on  a  single  determination  of  the 
chronometer  error  from  altitudes  on  one  side  of  the  meridian, 
it  is  better  to  observe  the  same  body  on  both  sides  of  the 
meridian,  and,  if  possible,  at  about  the  same  altitude.     The 
error  of  the  chronometer  having  been  found  from  each  set  of 
sights,  the  mean  is  taken  as  the  correct  error,  and  this  mean 
will  probably  be  nearer  the  true  error  than  the  result  from 
either  set ;  the  effect  of  the  constant  errors  of  latitude,  instru- 
ment, and  observer,  being  opposite  in  the  two  cases,  will  be 
eliminated  by  taking  the  mean. 

(c)  By  Equal  Altitudes;  Deduction  of  the  Equation  of 
Equal  Altitudes. 

269.  If  a  heavenly  body  is  observed  at  a  given  altitude  on 
one  side  of  the  meridian,  and,  again,  at  the  same  altitude  on 
the  other  side  of  the  meridian,  the  chronometer  times  of  each 
observation  being  noted,  the  mean  of  the  two,  or  the  middle 
chronometer  time,  will  be  the  time  of  the  body's  transit,  pro- 
vided its  declination  has  not  changed  in  the  interval.     The 
difference  between  this  C.  T.  of  transit  and  the  actual  time  of 
transit,  found  independently,  will  be  the  chronometer  error  on 
that  particular  time. 

Fixed  stars  are  bodies  whose  change  of  declination  is  so 
slight  that  it  may  be  neglected,  and  to  these  the  above  remarks 
apply. 

For  a  body  whose  declination  changes  in  the  interval  be- 
tween observations,  the  hour  angles  at  the  same  altitude,  East 
and  West  of  the  meridian,  are  not  numerically  equal,  the 
East  or  West  hour  angle  being  the  larger  according  to  cir- 
cumstances, the  difference  being  due  to  the  change  of  declina- 
tion in  the  interval.  Half  the  difference  of  these  hour 
angles,  called  the  "  Equation  of  Equal  Altitudes/'  is  the  hour 


564 


NAUTICAL  ASTRONOMY 


angle  of  the  body  at  the  middle  chronometer  time ;  or,  in  other 
words,  the  correction  to  be  applied  to  the  middle  chronometer 
time  to  obtain  the  chronometer  time  when  the  body's  center 
is  on  the  local  meridian. 

Let  Fig.  119  represent  a  projection  on  the  plane  of  the 
horizon.  NS  is  the  meridian.  EQW  is  the  equator.  PM', 
PM",  PM,  and  Pm  are  hour  circles.  MnM"Mf  is  a  parallel 


FIG.  119. 

of  altitude.  dMcM'w  is'  the  diurnal  circle  of  a  star  or  body 
whose  declination  does  not  change,  so  that  if  the  altitude  of 
the  star  is  observed  at  M,  East  of  the  meridian,  and  again  at 
M ',  when  it  is  on  the  same  parallel  of  altitude  West  of  the 
meridian,  the  hour  angle  M'PZ  equals  M'PZ,  and  if  the  times 
be  noted  when  the  star  is  at  M  and  M',  the  mean  of  these, 
ignoring  the  rate  of  the  time  piece,  will  be  the  time  of  tran- 
sit at  c.  When  the  star  is  at  c  on  the  upper  branch  of  the 
meridian,  its  E.  A.  is  the  L.  S.  T.  Knowing  the  longitude, 
this  L.  S.  T.  can  be  converted  into  G-.  M.  T.,  the  difference 


EQUAL  ALTITUDES  566 

between  which  and  the  chronometer  time  of  transit  will  be 
the  error  of  the  chronometer  on  G.  M.  T. 

If  the  star  is  West  of  the  meridian  at  the  first  observation, 
the  mean  of  the  times  will  correspond  to  the  instant  when  the 
star  is  on  the  lower  branch  of  the  meridian,  at  which  time 
the  L.  S.  T.  =  R.  A.  of  the  star  plus  12  hours,  and  this  L. 
S.  T.  can  be  converted  into  Gr.  M.  T.,  and  the  error  of  chro- 
nometer found  as  above  explained. 

If  the  heavenly  body  be  one,  as  the  sun  for  instance,  whose 
declination  changes  during  the  interval  between  the  obser- 
vations, the  hour  angles  at  the  same  altitudes  East  and  West 
of  the  meridian  will  not  be  numerically  equal. 

In  the  figure,  a  case  is  assumed  in  which  the  declination  of 
the  sun  is  of  the  same  name  and  less  than  the  latitude,  and  the 
sun  is  moving  toward  the  elevated  pole.  If  the  sun  is  ob- 
served when  it  is  at  M,  its  altitude  being  M E,  in  its  diurnal 
path  it  will  not  follow  dew,  but  will  follow  the  circle  dmM" 
and  will  reach  the  parallel  of  altitudes  at  M"  instead  of  at  M'. 
The  western  hour  angle  will  evidently  differ  from  the  eastern 
one  by  M"PM'.  This,  then,  is  the  change  in  the  western  hour 
angle  during  the  local  apparent  time  represented  by  the  east- 
ern and  western  hour  angles  if  and  t",  due  to  the  total  change 
in  declination  in  those  times.  If  the  times  had  been  noted 
by  a  mean  time  chronometer  when  the  sun  was  at  M  and  M", 
the  mean  of  these  times,  ignoring  as  before  the  rate  and  in 
addition  the  change  in  the  equation  of  time  during  the  inter- 
val, will  correspond  not  to  the  time  of  transit  but  to  the  in- 
stant when  the  sun  is  at  ra  and  when  its  hour  angle  is'mPZ, 
which  is  clearly  equal  to  one-half  M"PM' ';  or,  in  other  words, 
one-half  the  change  in  the  western  hour  angle  during  the  in- 
terval between  the  observations,  due  to  a  change  in  the  decli- 
nation. 

If  this  small  angle  be  reduced  to  the  same  unit  as  that  of 
the  chronometer  and  applied  to  the  mean  of  the  chronometer 
times,  the  result  will  be  the  chronometer  time  of  transit  or 


566  NAUTICAL  ASTRONOMY 

local  apparent  noon.  If  the  error  in  the  western  hour  angle  is 
positive  or  increases  that  angle,  the  small  angle  mPZ  is  to  the 
westward  of  the  meridian  and  its  sign  of  application  as  a  cor- 
rection is  minus.  If  the  error  is  negative  or  decreases  the 
western  hour  angle,  its  sign  as  a  correction  is  plus ;  or,  in  other 
words,  its  sign  of  application  is  opposite  from  that  gotten  by 
differentiation. 
Let  L  be  the  latitude,  always  plus ; 

d,  the  declination  of  sun,  plus  if  of  same  name  as  L; 
dd,  the  hourly  change  of  decimation  at  L.  A.  noon,  plus 

if  body  is  moving  toward  the  elevated  pole; 
f,  the  eastern  hour  angle; 
t",  the  western  hour  angle ; 
dt",  the  hourly  change  in  H.  A.  due  to  dd  in  western 

H.  A.; 

2t,  the  elapsed  time  by  chronometer. 

As  M'PM"  is  the  error  produced  in  the  western  H.  A.  by 
the  entire  change  in  declination  between  observations,  we  as- 
sume the  declination  to  vary  uniformly  and  regard  d  and  t  as 
variable  in  the  general  equation  for  the  western  H.  A.,  t". 
sin  Ji  =sin  L  sin  d  -f-  cos  L  cos  d  cos  t", 
0  =  sin  L  cos  d  dd  —  cos  L  cos  t"  sin  d  dd 

—  cos  L  cos  d  sin  t"dt", 
„  _        dd  tan  L      dd  tan  d 
sin*"        "Ta¥7'~ 

Since  dd  is  taken  as  the  hourly  change  of  declination  at 
the  approximate  middle  instant,  in  other  words,  at  L.  A.  noon, 
dt"  is  the  change  in  the  H.  A.  in  one  hour,  but  the  complete 
change  takes  place  in  L.  A.  times  t'  and  t",  expressed  in 
hours;  therefore,  the  angle  M'PM"  is 

ft*  -L  r>\  w  -  A.  <&*(*'  +  *")  tan  £      <M(?  +  ntand 

Bin*"  tan  t" 

The  hours  (tr  +  O  are  apparent  time,  but  it  is  more  con- 
venient to  use  the  elapsed  time  (2£),  as  shown  by  chronometer 
between  observations,  and  to  substitute  t  for  t"  or  t' .  This  in- 


EQUAL  ALTITUDES  567 

volves  a  practically  inappreciable  error  due  to  change  in  equa- 
tion of  time  and  in  rate  of  chronometer  during  interval; 
hence  we  have 

2rf  dd  tan  L       2tdd  tan  d 

%tdt= .     ,  —2 

sin  t  tan  t 

As  this  error  has  been  shown  to  be  twice  the  H.  A.,  mPZ, 
at  the  middle  instant,  we  divide  the  above  by  2,  and,  to  ex- 
press it  in  time,  divide  the  right-hand  member  by  15,  and  as 
this  is  an  error  whose  sign  of  application  as  a  correction  has 
been  shown  to  be  the  reverse,  we  change  the  signs  of  the  right- 
hand  member  and  have 

fjf  tdd  tan  L  ,   tdd  tan  d  /oo*\ 

15  sin  t    +  15  tan  t  ' 
which  is  the  equation  of  equal  altitudes. 

Rules. 

L  is  always  positive. 

d  is  positive  if  of  same  name  as  latitude  or  of  elevated  pole. 

dd  is  positive  if  body  is  moving  towards  elevated  pole. 

t  is  always  positive. 

tdt  in  seconds  is  applied  with  its  proper  sign  to  the  middle 
chronometer  time,  giving  chronometer  time  of  L.  A.  noon, 
or  error  of  chronometer  fast  on  L.  A.  T. 

To  this,  the  equation  of  time  is  applied  as  to  mean  time, 
giving  chronometer  time  of  local  mean  noon,  or  error  of  chro- 
nometer fast  on  L.  M.  T.  at  L.  A.  noon. 

To  this  is  applied  the  longitude,  adding  when  East,  sub- 
tracting when  West,  giving  error  of  chronometer  fast  of 
G.  M.  T.  at  L.  A.  noon. 

d,  dd,  and  equation  of  time  are  taken  from  the  Nautical  Al- 
manac for  the  instant  of  local  apparent  noon.  The  hourly 
change  of  the  equation  of  time  and  declination  are  corrected 
for  second  differences. 

In  p.  m.  and  a.  m.  equal  altitudes,  it  is  evident  that  when 
equal  zenith  distances  are  observed  in  a  latitude  L,  their  sup- 


568  NAUTICAL  ASTRONOMY 

plements  may  be  considered  as  equal  zenith  distances  observed 
at  the  antipode  in  latitude  -  -  L  on  the  same  meridian. 
Hence  the  formula  will  give  the  equation  for  noon  at  the 
antipode,  or  for  midnight  at  the  place  of  the  observer,  by  sub- 
stituting —  L  for  L  in  the  first  term  of  the  equation,  which, 
therefore,  becomes 

t  dd  tan  L    .    t  dd  tan  d 

t  dt  —  -\ =- — : — 2 — f-    _g  , — 3-.  (227) 

15  sm  t  15  tan  t 

In  p.  m.  and  a.  m.  equal  altitudes  of  the  sun,  or,  in  other 
words,  when  the  first  observation  is  West  of  the  meridian, 
the  "equation  of  equal  altitudes,"  tdt,  is  applied  with  its 
proper  sign  to  the  middle  chronometer  time,  giving  the  chro- 
nometer time  of  local  apparent  midnight. 

To  this,  the  equation  of  time  is  applied  as  to  mean  time, 
giving  the  chronometer  time  of  local  mean  midnight,  or  error 
of  chronometer  fast  on  L.  M.  T.  at  local  apparent  midnight. 
To  the  C.  T.  of  local  mean  midnight  is  applied  the  longitude, 
adding  when  East,  subtracting  when  West;  the  result  is  the 
error  of  chronometer  fast  of  G.M.T.  at  local  apparent  midnight. 

d,  dd,  and  the  equation  of  time  are  taken  from  the  Nauti- 
cal Almanac  for  the  instant  of  local  apparent  midnight  (12 
hours  -f-  A). 

The  hourly  change  of  equation  of  time  and  of  declination 
are  corrected  for  second  differences. 

Ex.  206.— April  4,  1905,  p.  m.,  in  latitude  38°  59'  N., 
longitude  5h  05m  563.5  W.,  observed  equal  altitudes  of  a  Leonis 
(Eegulus),  noting  time  by  a  mean  time  chronometer.  C.  T.  * 
East  of  meridian  llh  04m  26s.  C.  T.  *  West  of  meridian  3h 
32m  30s.  Find  the  C.  C.  on  G.  M.  T. 


C.  T.  *  East  of  mer. 
C.  T.  *  West  of  mer. 
Sum 
Middle  C.  T. 
G.  M.  T.  of  transit 
Chro.  slow  of  G.  M.  T. 

h  m     s 
11  0426 
332  30 

R.  A.  M.0  Gr.  M.  N. 
Red.  for  A  Table  II  I 
Sid.  time  local  Oh 
L.  S.  T.  =  *'sR.A. 
Sid.  int.  from  noon 
Red.  Table  II 
L.  M.  T.  of  transit 
Longitude  West 
G.  M.  T.  of  transit 

h  m     s 
April  4,    0  48  31.69 
+         50.258 

26  36  56 
1  18  28 
14  18  23.808 

0  49  21.948 
10  03  20.01 
9  13  58.062 
-      1  30.754 
9  12  27.306 
5  05  56.5 

0  59  55.808 

14  18  23.808 

EQUAL  ALTITUDES 


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572  NAUTICAL  ASTRONOMY 


Chauvenet's  Tables.—  If  in  formula  (226)  we  let 

A  =  —  :—  —  :  —  -  and  B  =  —-—  -  -,  that  formula  reduces  to 
15  sin  t  15  tan  t 

tdt  =  A.dd.i^L+B.dd.  tan  A 

and  formula  (227)  becomes 


Chauvenet  has  tabulated  log  A  and  log  B  (to  the  fourth  place 
only),  with  their  proper  signs  for  noon  and  midnight  transits, 
the  argument  being  2t.  These  logs  are  embodied  in  Table  37, 
Bowditch.  Particular  attention  must  be  paid  to  the  signs. 
However,  the  advantage  of  using  these  tables  is  not  apparent. 

The  equation  of  time  is  applied  to  the  C.  T.  A.  N.  (or  C. 
T.  A.  M.)  with  the  sign  of  application  to  mean  time,  because, 
if  it  is  -f-  to  mean  time,  apparent  time  is  greater  than 
mean  time,  the  apparent  sun  crosses  the  meridian  first,  and 
the  C.  T.  of  mean  noon  (or  midnight)  is  later  than  the  C.  T. 
of  apparent  noon  (or  midnight).  If  the  sign  of  application 
is  (  —  )  to  mean  time,  apparent  time  is  less  than  mean  time, 
the  apparent  sun  crosses  the  meridian  after  the  mean  sun, 
and  the  C.  T.  of  mean  noon  (or  midnight)  is  less  than  the 
C.  T.  of  apparent  noon  (or  midnight). 

To  the  C.  T.  of  mean  noon,  East  longitude  in  time  is  added 
because  the  Greenwich  mean  time  of  Greenwich  mean  noon  is 
desired,  which  will  not  occur  till  the  number  of  hours  repre- 
senting the  longitude  have  elapsed  since  local  mean  noon, 
when  the  longitude  is  East.  To  the  C.  T.  of  mean  noon, 
West  longitude  is  subtractive  because  Greenwich  noon  occurs 
before  noon  at  places  in  West  longitude  by  that  number  of 
hours. 

For  similar  reasons,  longitude  is  applied  to  the  C.  T.  of 
mean  midnight;  adding  when  East,  subtracting  when  West. 

Attention  is  called  to  the  fact  that  the  elapsed  time,  to  be 


EQUAL  ALTITUDES  573 

exact,  should  be  corrected  for  the  rate  of  the  chronometer 
and  also  for  the  change  in  the  equation  of  time  during  the 
interval;  but  such  refinement  is  usually  neglected,  being  of 
no  practical  importance,  except  when  the  chronometer  has  a 
large  rate. 

In  case  of  planets. — When  working  equal  altitudes  of 
planets,  formulae  (226)  and  (227)  will  apply  provided  2t 
may  be  expressed  in  units  of  the  hour  angle  of  the  body  ob- 
served instead  of  in  mean  time  units,  2t  being  the  sum  of  the 
hour  angles  of  the  body  itself.  Therefore, 

let  2t  =  the  sum  of  the  two  hour  angles  of  the  planet; 

2s  =  the  elapsed  sidereal  time  between  the  observations ; 
2tm  =  the  elapsed  mean  time  between  the  observations; 
dr  =  the  mean  change  in  the  planet's  right  ascension  in 
one  hour  of  mean  time,  expressed  in  decimals 
of  an  hour. 

Then  2tmdr  will  be  the  total  change  in  the  planet's  right 
ascension  in  2tm,  or  the  elapsed  time  by  chronometer,  and  the 
elapsed  sidereal  time  will  be  greater  than  the  sum  of  the 
hour  angles  by  this  total  change;  or, 

2s  =  2t  +  2tmdrf  or  s  =  t  +  tmdr.  (230) 

From  (137),  s  =  tm  +  .0027379  tm; 

therefore,        t  =  tm  +  tm  (.0027379  —  dr) .  (231) 

Again,  if  dd  represents,  as  in  the  formulae  (226)  and  (227), 
the  hourly  change  of  the  planet's  declination  at  the  instant  of 
transit  for  one  of  its  own  hours,  the  total  change  in  declina- 
tion will  be 

2t.dd.  =  [2tm  +  2tm  (.0027379  —  dr)]dd.; 

and,  if  ddm  represents  the  change  in  the  planef  s  declination 
for  one  hour  of  mean  time  at  the  instant  of  the  planet's  tran- 
sit, the  total  change  will  be  2tmddm;  therefore, 

[2tm  +  2tm  (.0027379  —  dr)]  dd  =  2tmddm;  or 

<^  =  170027379 -~rfr* 


574  NAUTICAL  ASTRONOMY 

A 

270.  Littlehales'  method  of  equal  altitudes.  —  Equation  226 
may  be  placed  under  the  form 

,   _    —  tdd  /sin  L  cos  d  —  cos  L  sin  d  cos  t  \ 
15    \  cos  L  cos  d  sin  t  ) 

but  sin  L  cos  d  —  cos  L  sin  d  cos  t  =  cos  /z,  cos  M,  and 

cos  fc      __      1 
cos  L  sin  £       sin  M 

therefore  tdt  =  ^y^-  sec  d  cot  M.  (226a) 

d  and  dd  are  taken  from  the  Almanac  for  local  apparent  noon 
and  marked  as  explained  in  Art.  269;  then  tdt  found  from 
(226a)  is  applied  in  finding  chronometer  error  as  explained 
in  the  same  article,  the  form  of  arrangement  of  work  being 
modified  only  in  the  logarithmic  column. 

To  illustrate  the  process  let  us  refer  to  Ex.  207  in  which 
Lat.  =  33°  51'  41"  S.,  d  —  5°  45'  52"  N.,  dd  =  N.  57".135, 
and  t  =  3h.3764. 

Entering  the  Azimuth  Tables,  *<UVRO"N>    SPP          4- 

=^™  cot          + 


,  = 

n  latitude  column,  ,,      ,._,,  10KXT       , 

and  S*  37-  or  12*  -«  (since  eMs          "*%%£?*  °g 

-),wefindJf=46°35';there-             <  =  f-3764  log 

for'e  by  substitution  in  (226a),          .  ..      1512s  33  "Ol°g 

we  find  tdt  as  shown  in  the  col-  *«Z!?3-*S  g 
umns  to  right. 

The  sign  of  application  of  tdt  to  the  middle  chronometer 
time  may  be  found  as  above  by  following  the  signs  of  the 
quantities  involved,,  or  by  using  the  following  simple  precept  : 
"  For  values  of  the  position  angle  less  than  90°  ,  tdt  should  be 
added  when  the  polar  distance  is  increasing  and  subtracted 
when  the  polar  distance  is  decreasing;  and  for  values  of  the 
position  angle  greater  than  90°  ,  the  reverse  is  the  case/' 

When  the  first  observation  is  west  of  the  meridian.  —  It  is 
evident  when  equal  zenith  distances  are  observed  in  a  latitude 
L,  in  this  case,  that  their  supplements  may  be  considered  as 
equal  zenith  distances  observed  at  the  antipode  in  latitude  —  L 
on  the  same  meridian;  and  that  in  the  triangle  to  be  con- 
sidered, which  includes  the  elevated  pole  and  the  antipode, 
the  angle  at  the  body  will  be  180°  —  M  instead  of  M.  Hence, 
we  shall  obtain  the  equation  for  noon  at  the  antipode  or  for 


EQUAL  ALTITUDES  575 

midnight  at  the  place  of  observer  by  substituting  180°  —  M 
for  M  in  (226a)  ; 


therefore,  tdt  =       r-  sec  d  cot  M.  (ma) 

d  and  dd  are  taken  from  the  Almanac  for  local  apparent  mid- 
night and  marked  as  in  Art.  269.  The  sign  of  application  of 
tdt  to  the  middle  C.  T.  is  found  by  following  the  signs  of  the 
quantities  involved  in  (227a),  or,  by  a  reverse  application  of 
the  precept  given  for  a.  m.  and  p.  m.  observations.  When 
taking  out  M  from  the  azimuth  tables,  it  must  not  be  forgot- 
ten that  the  hour  angle  from  the  upper  meridian  in  this  case 
is  taken  as  the  supplement  of  half  the  elapsed  time  or  12h  —  t. 
(See  Appendix  C.) 

271.  To  correct  the  middle  time  for  a  small  difference  of 
altitude. 

The  altitude  at  the  second  observation  may  differ  slightly 
from  that  at  the  first  observation  through  a  change  in  refrac- 
tion which  may  be  learned  by  noting  the  barometer  and  ther- 
mometer during  both  observations;  through  a  change  in  the 
index  correction;  or  through  interference  of  clouds  or  other 
unexpected  causes. 

-.  coshdh  (  dh  in  arc,   /rtOOX 

From  Art.  237,  dt  =     -  15  cos  £  cos  d  sin  ;  {  dt  in  tim;.  (233) 

In  this  formula,  dh  is  negative  when  dt  is  positive,  since 
as  hour  angle  increases,  altitude  decreases  and  vice  versa. 

If  dh  represents  the  difference  between  the  altitude  ob- 
served, and  the  one  that  should  have  been  observed,  dt  will  be 
the  corresponding  difference  in  hour  angles.  This  being  the 
change  during  the  whole  elapsed  time,  \dt  will  be  the  correc- 
tion to  be  added  to  the  middle  chronometer  time  when  the 
western  .  altitude  is  the  greater  ;  to  be  subtracted,  when  the 
western  altitude  is  the  smaller. 

Or,  if  desired,  take  the  difference  between  two  readings  of 
the  sextant  representing  double  angles  by  artificial  horizon, 
and  the  difference  of  corresponding  times.  Find  the  change 
in  time  due  to  1'  or  1"  of  double  altitude  and  multiply  it  by 
the  known  inequality  of  altitudes.  This  result  will  be  the 
correction  to  the  middle  chronometer  time,  to  be  added  when 
the  western  altitude  is  the  greater  ;  otherwise,  subtracted. 


576 


NAUTICAL  ASTRONOMY 


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METHOD  OF  OBSERVATION  577 

Time  and  method  of  observation. — The  most  favorable  posi- 
tion of  a  heavenly  body  for  observations  for  equal  altitudes 
is  when  the  body  is  on  the  prime  vertical ;  small  errors  in  alti- 
tude having  the  least  effect  on  the  resulting  hour  angle  at  that 
time,  though  the  altitude  should  be  sufficiently  great  to  elimi- 
nate errors  of  refraction,  at  all  events  to  lessen  the  probabili- 
ties of  them.  For  preparing  the  artificial  horizon  see  Art.  154. 

The  altitudes  of  the  same  limb  should  be  observed  at  reg- 
ular intervals  of  10'  or  20'  of  arc  with  the  artificial  hori- 
zon; as  soon  as  contact  is  made  at  one  division,  set  the 
sextant  at  the  next,  watch  for  contact,  and  mark  the  time 
at  each  contact.  In  forenoon  single  altitudes  the  following 
method  may  be  used:  having  observed  the  2  TJ  at  intervals 
of  10'  of  arc  through  50'  of  a  given  degree,  run  the  instru- 
ment arm  back  and  observe  the  2  jQ.  at  intervals  of  10'  up  to 
50'  through  the  same  degree.  The  mean  of  these  will  be 
the  sextant  altitude  2  -0-,  eliminating  correction  for  S.  D. 
The  reverse  procedure  should  be  followed  for  afternoon  single 
altitudes.  The  corresponding  sets  of  altitudes  on  each  side 
of  the  meridian  should  be  taken  under  like  conditions,  with 
the  same  instrument  and  adjustments  and  with  the  same 
end  of  the  roof  toward  the  observer.  In  equal  altitudes  it 
is  not  necessary  that  the  exact  altitude  at  either  observa- 
tion be  known,  but  only  the  times  at  which  the  altitudes 
East  and  West  of  the  meridian  are  equal,  since  the 
elapsed  time  alone,  and  not  the  altitude,  is  required  for  the 
computation. 

272.  Comparison  of  methods. — In  equal  altitudes  of  stars, 
latitude  and  declination  do  not  enter,  and  hence  errors  in  them 
do  not  affect  the  result. 

The  mean  of  errors  by  double  altitudes  of  the  sun,  at  about 
the  same  altitudes  East  and  West  of  the  meridian,  will  be 


578 


NAUTICAL  ASTRONOMY 


practically  the  same  as  the  error  by  equal  altitudes.  In  high 
latitudes,  the  double  altitude  method  is  preferable  on  account 
of  the  large  value  of  tan  L,  in  the  equation  of  equal  altitudes. 

LONGITUDE. 

273.  Longitude  has  been  denned  as  the  difference  in  the 
hour  angles  of  the  same  heavenly  body  at  a  given  instant  at  the 
local  and  prime  meridians ;  the  prime  meridian  being  usually 
that  of  Greenwich,  and  is  marked  East  when  the  local  time  is 
the  greater,  West  when  the  Greenwich  time  is  the  greater. 

In  Fig.  120,  let  'PG  be  the  meridian  of  Greenwich;  PM  the 
local  meridian  West  of  Greenwich;  PM'  the  meridian  of  a 

place  East  of  Greenwich;  PS  the 
declination  or  hour  circle  of  a 
heavenly  body;  P  T  the  hour 
circle  passing  through  the  vernal 
equinox.  Then  GPS  is  the  Green- 
wich hour  angle  of  the  body  8; 
MPS  its  hour  angle  at  all  places 
on  the  meridian  PM;  M'PS  its 
hour  angle  at  all  places  on  the 
meridian  PM'.  GPM  is  the  lon- 
gitude of  the  meridian  PM  and  is  marked  West.  GPM'  is  the 
longitude  of  the  meridian  PM'  and  is  marked  East. 


FIG.  120. 


=  GPS  — MPS,  \ 
=  M'PS— GPS.) 


GPM  =  GPS  — MPS, 
also  GPM' 


(234) 


If  PS  is  the  true  sun's  hour  circle,  GPM  or  GPM'  is  the 
difference  of  apparent  times  for  the  same  instant  at  the  Green- 
wich and  local  meridians ;  if  it  is  that  of  the  mean  sun,  GPM 
or  GPM'  is  then  the  corresponding  difference  of  mean  times. 
Again, 

GPM  =  OPT— MPT,  1 
and  GPM'  =  M'PT  —  0PT.  J 


LONGITUDE  579 

or  the  longitude  of  a  place  is  the  difference  between  the 
Greenwich  and  local  sidereal  times  at  the  same  instant. 

The  problem,  then,  of  finding  the  longitude  of  a  place  con- 
sists in  determining  for  the  same  instant  of  time  the  differ- 
ence of  the  Greenwich  and  local  times,  apparent,  mean,  or 
sidereal. 

When  the  navigator  knows  the  error  and  rate  of  his  chro- 
nometer, found  at  a  place  of  known  longitude,  he  can  find  for 
the  instant  of  observation  at  the  place  the  Greenwich  mean 
time  corresponding.  This  Greenwich  mean  time  may  be 
converted  into  Greenwich  apparent  time,  in  case  the  sun  is 
the  body  observed,  by  applying  the  equation  of  time ;  or,  into 
Greenwich  sidereal  time  (Art.  192),  if  the  body  observed  is 
the  moon,  a  planet,  or  a  star. 

The  problem  of  finding  the  hour  angle  of  a  heavenly  body 
has  been  considered  in  Art.  226. 

In  case  the  body  observed  is  the  sun,  the  hour  angle  reck- 
oned positively  to  the  West  up  to  24  hours  is  the  local  appar- 
ent time  (astronomically  considered) ;  the  difference  between 
which  and  the  G.  A.  T.  found  above  will  be  the  longitude,  East 
when  the  L.  A.  T.  is  greater  than  the  G.  A.  T.,  otherwise 
West. 

In  case  the  body  observed  is  any  other  heavenly  body,  solve 
the  astronomical  triangle  for  t,  the  hour  angle,  mark  it  + 
when  the  body  is  West  of  the  meridian,  ( — )  when  it  is  East 
of  the  meridian.  The  algebraic  sum  of  the  body's  hour  angle 
and  right  ascension  will  be  the  local  sidereal  time ;  the  differ- 
ence between  which  and  the  G.  S.  T.  found  above  will  be  the 
longitude,  East  when  the  L.  S.  T.  is  the  greater,  West  when 
the  G.  S.  T.  is  the  greater  (see  Art.  179). 


580  NAUTICAL  ASTRONOMY 

Determination  of  Longitude. 

274.  Longitude  may  be  determined 

(a)  Ashore,  by  electric  signals; 

(b)  Ashore  .or  afloat, 

(1)  By  equal  altitudes, 

(2)  By  single  or  double  altitudes. 

(a)  Ashore. 

275.  By  electric  signals. — For  this  method  there  should  be 
observatories,  permanent  or  portable,  in  telegraphic  communi- 
cation with  each  other.     Each  observatory  should  be  provided 
with  a  transit  instrument,  an  electro-chronograph,  and  requi- 
site telegraphic  instruments.  An  astronomical  clock  of  known 
error  and  rate  should  be  at  one  station,  or  it  may  be  at  any 
place  which  is  in  telegraphic  communication  with  both  sta- 
tions.    The  electric  connections  should  be  such  that  during 
the  times  of  observations  each  successive  beat  of  the  astro- 
nomical clock  will  be  recorded  simultaneously  on  the  chrono- 
graph sheet  at  each  station,  as  well  as  the  instant  of  transit 
across  each  meridian  of  a  given  star  previously  agreed  upon; 
the  clock  beats  being  recorded  by  connection  through  the  pen- 
dulum, and  the  times  of  transit  through  a  signal  key  in  the 
hands  of  the  observers.     The  intervals  between  two  successive 
beats  of  the  clock,  as  recorded  on  the  chronograph  sheet,  being 
subdivided  by  scale,  the  instants  of  transit  of  the  star  across 
both  eastern  and  western  meridians,  and  hence  the  elapsed 
time  between  transits,  may  be  obtained  to  fractions  of  a  sec- 
ond from  each  chronograph. 

The  difference  between  the  times  of  transit  of  the  same  star 
across  the  two  meridians,  as  indicated  on  each  chronograph, 
is  corrected  for  the  rate  of  the  clock  in  the  interval,  and  the 
mean  of  the  two  values,  thus  corrected,  is  taken  as  the  dif- 
ference in  longitude  required. 


BY  ELECTRIC  SIGNALS  581 

Another  method  is  to  have  at  each  station  a  number  of 
break  circuit  chronometers,  mean  or  sidereal,  instead  of  the 
astronomical  clock  at  one  station.  Sidereal  chronometers 
are  preferred  as  they  can  be  rated  by  star  observations  with 
less  computation. 

The  chronometers  at  each  station,  having  been  carefully 
rated  for  local  time  by  transits  of  stars,  are  compared  by 
signals  sent  first  one  way,  then  the  other  way;  the  times  of 
sending  and  receiving  signals  being  recorded  at  both  stations. 
The  readings  of  the  chronometers  at  both  stations  are  reduced 
to  local  time,  and,  if  the  signals  are  recorded  simultaneously 
at  both  stations,  the  difference  of  the  local  times  at  the  instant 
of  comparison  will  be  the  difference  of  longitude.  In  other 
words,  if  Te  is  the  local  time  of  that  instant  at  the  eastern 
station  A  and  Tw  the  corresponding  local  time  at  the  western 
station  B,  then  D  —  Te  —  Tw. 

However,  the  record  of  signals  is  not  simultaneous,  time  is 
lost  in  completing  the  circuit  and  in  the  action  of  the  arma- 
ture; this  lost  time,  called  the  "wave  and  armature  time/' 
may  be  represented  by  x.  Therefore,  if  the  signal  is  sent 
from  A,  the  time  recorded  at  B  will  not  be  Tw  but  will  be 
Tw  +  x>  and  the  difference  of  times  represented  by  iy  will  be 
D'  =  Te—(Tw  +  x). 

If,  however,  the  signal  is  sent  to  A  by  the  observer  at  B 
at  the  local  time  Tw  of  the  western  station,  the  corresponding 
time  recorded  at  A  will  not  be  Te  but  will  be  Te  +  x  and  the 
difference  of  times  represented  by  D"  will  be 

—  Tw, 


„,  m        ^'    ^ 

=  Te  —  Tw  which  equals  D. 

A  .  </ 

It  is  thus  seen  that  if  we  take  the  difference  of  longitude 
to  be  the  difference  of  the  local  times  indicated  at  the  instant 
of  comparison  when  the  signal  is  sent  from  the  eastern  to  the 


582  NAUTICAL  ASTRONOMY 

western  station,  the  difference  of  longitude  will  be  too  smaL 
by  a  fraction  of  a  second;  and,  in  the  same  way,  it  would  be 
too  large  by  the  same  amount  when  the  signal  is  sent  from 
the  western  to  the  eastern  station ;  the  error  being  eliminated, 
however,  by  transmitting  signals  both  ways  and  taking  the 
mean  of  the  two  results  for  the  correct  difference  of  longi- 
tude. 

(b)  Ashore  or  Afloat. 

276.  Longitude  by  equal  altitudes. — If  equal  altitudes  of 
a  heavenly  body  be  observed  East  and  West  of  the  meridian, 
and  the  times  noted  by  a  chronometer,  or  by  a  watch  com- 
pared with  a  chronometer,  the  mean  of  the  chronometer  times 
will  be  the  chronometer  time  of  its  meridian  transit,  provided 
the  observer  has  not  changed  his  position,  or  the  body  its  de- 
clination, in  the  interval.  The  known  chronometer  correction 
having  been  applied  to  this  middle  chronometer  time,  the 
result  will  be  the  Greenwich  mean  time  of  transit.  If  the 
observation  is  made  ashore,  one  condition  is  fulfilled;  if  the 
body  observed  is  a  star,  the  other  condition  is  fulfilled.  The 
star's  right  ascension  will  be  the  L.  S.  T.  of  transit  and 
knowing  the  G.  M.  T.  of  transit,  the  L.  M.  T.,  and  then 
the  longitude,  may  be  found. 

If  the  declination  of  the  body  has  changed  in  the  interval 
between  observations,  as  may  be  expected  in  the  case  of  the 
sun,  the  moon,  or  a  planet,  the  correction  to  the  middle  Q-. 
M.  T.  of  observation  for  such  a  change  must  be  ascertained 
and  properly  applied  to  find  the  Gr.  M.  T.  of  transit. 

Observations  for  time  of  the  moon  and  planets  being  un- 
desirable for  reasons  stated  in  Art.  231,  consideration  of  the 
sun  alone  comes  under  this  article. 

The  general  method  of  finding  the  correction  to  the  middle 
time  due  to  a  change  of  declination  in  the  interval  t,  or  the 
value  of  tdt,  pursued  in  Art.  269,  when  working  for  chro- 


LONGITUDE  BY  EQUAL  ALTITUDES  583 

nometer  error,  may  be  followed  when  working  for  longitude, 
with  certain  essential  changes  in  the  method. 

When  working  for  chronometer  error,  the  longitude  must 
be  known,  and  in  the  case  of  the  sun,  the  values  of  the  decli- 
nation, equation  of  time,  and  dd  are  easily  found  for  the 
instant  desired,  either  of  local  apparent  noon  or  midnight. 
However,  when  working  equal  altitudes  for  longitude,  the 
nearest  known  time  to  the  instant  of  the  sun's  transit  is  the 
G-.  M.  T.  of  the  middle  instant  between  observations. 

Take  from  the  Nautical  Almanac  for  this  G-.  M.  T.  the 
declination,  equation  of  time,  and  dd,  though  it  must  not  be 
forgotten  that,  strictly  speaking,  they  should  be  for  the  instant 
of  transit,  an  unknown  time  in  this  case.  In  other  words, 
follow  the  methods  of  Ex.  208,  Art.  269,  in  correcting  the  de- 
clination and  equation  of  time  and  finding  dd,  but  substi- 
tuting G.  M.  T.  of  the  middle  instant  for  longitude. 

Then,  having  found  from  the  equation  of  equal  altitudes 
(226)  or  (227),  according  as  the  first  observation  of  the 
sun  was  East  or  West  of  the  meridian,  the  value  of  tdt,  and 
having  the  equation  of  time,  we  would  have  the  following 
form  for  the  arrangement  of  the  work: 
G.  M.  T.  of  middle  instant 

tdt  =  ± 

G.  M.  T.  of  local  apparent  noon  or  midnight  "| 

(according  as  first  observation  was  E.  or  V  = 

W.  of  meridian). 

Eq.  of  time  (sign  of  application  to  M.  T.)  

G.  A.  T.  of  local  apparent  noon  or  midnight  ^ 

(according  as  first  observation  was  E.  or  \= 

W.  of  meridian). 

The  G.  A.  T.  of  noon,  if  less  than  12  hours,  is  the  longi- 
tude West;  if  greater  than  12  hours,  take  it  from  24  hours 
and  the  remainder  is  the  longitude  East. 


584  NAUTICAL  ASTRONOMY 

From  the  G.  A.  T.  of  midnight  subtract  12  hours;  the  re- 
mainder, if  plus,  is  the  longitude  West;  if  minus,  it  is  the 
longitude  East. 

The  declination  of  the  sun,  the  H.  D.  of  declination,  and 
the  equation  of  time,  when  working  for  longitude,  are  taken 
out  from  Page  II,  N.  A.,  for  the  middle  G.  M.  T.,  though 
strictly  speaking,  they  should  be  taken  out  for  the  instant  of 
transit. 

In  case  the  value  of  tdt  is  of  such  a  magnitude  as  to  ap- 
preciably affect  those  quantities,  it  would  be  better,  after 
finding  the  G.  M.  T.  of  transit,  to  take  them  out  for  that 
time  and  repeat  the  process.  Such  refinement,  however,  will 
probably  be  unnecessary,  especially  when  the  first  observation 
is  East  of  the  meridian. 

277.  When  the  position  of  the  observer  changes,  as  it  does 
at  sea,  in  order  to  attain  any  approach  to  accuracy,  it  is  neces- 
sary that  the  elapsed  interval  should  be  small  and  also  that 
the  conditions  should  be  favorable  for  finding  time.  In  low 
latitudes,  especially  when  the  latitude  and  declination  are 
nearly  the  same,  and  the  observations  are  of  the  sun  taken 
within  a  few  minutes  of  noon,  the  vessel  nearly  stationary  or 
not  changing  her  latitude,  the  conditions  may  be  said  to  be 
favorable  for  the  following  simple  solution. 

(a)  Approximate  method  for  longitude  from  equal  altitudes 
of  the  sun. — jSTote  the  time  by  a  chronometer  of  known  error 
when  the  sun  is  at  the  same  altitude,  East  and  West  of  the 
meridian.  The  mean  of  these  chronometer  times  may  be 
taken  without  much  error  as  the  chronometer  time  of  local 
apparent  noon;  apply  the  chronometer  correction,  finding  the 
G.  M.  T.  of  local  apparent  noon;  reduce  this  to  G.  A.  T.  by 
applying  the  equation  of  time.  This  G-.  A.  T.  will  be  the 
longitude  West  from  Greenwich;  if  the  G.  A.  T.  is  greater 
than  12  hours,  subtract  it  from  24  hours,  the  remainder  will 
be  the  longitude  East  from  Greenwich. 


LONGITUDE  BY  EQUAL  ALTITUDES 


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586  NAUTICAL  ASTRONOMY 

Such  an  approximate  application  of  "  equal  altitudes "  is 
only  available  in  the  tropics  under  conditions  named.  The 
method  of  equal  altitudes  for  longitude  has  a  more  extended 
application  when  stars  are  used,  as  suitable  ones  can  be  found 
in  any  latitude. 

(b)  Method  of  equal  altitudes  for  longitude  when  the 
positions  of  ship  and  body  change. — When  the  body  observed 
is  on  or  near  the  prime  vertical  and  the  change  of  latitude  is 
small,  the  error  involved  through  neglecting  this  change  will 
be  small;  however,  if  it  is  desired  to  correct  for  change  of 
position,  it  may  be  done  very  closely  in  one  of  the  following 
ways: 

(1)  The  correction  may  be  made  approximately  by  reset- 
ting the  sextant  at  the  second  observation,  so  that  the  second 
altitude  will  be  increased  by  the  number  of  minutes  of  arc 
equal  to  the  number  of  sea  miles  in  the  difference  of  latitude, 
when  the  vessel  sails  toward  the  sun ;  or  decreased  in  the  same 
ratio  when  she  sails  away  from  the  sun.     The  mean  of  the 
times  of  observations  will  then  be  without  appreciable  error 
the  time  of  transit. 

(2)  The  mean  of  the  times  of  equal  altitudes  of  a  heavenly 
body  corresponds  to  the  time  of  the  maximum  altitude,  so 
that  if  we  find  the  hour  angle  of  the  sun  at  its  maximum  alti- 
tude (Art.  246),  that  is,  the  interval  of  time  between  maxi- 
mum altitude  and  meridian  passage,  and  apply  it  to  the  mean 
of  the  Greenwich  mean  times  of  observation,  we  will  have 
the  Greenwich  mean  time  of  local  apparent  noon.     Applying 
to  this  the  equation  of  time,  we  will  obtain  the  G.  A.  T.  of 
local  apparent  noon  or  longitude  West.   Should  this  be  greater 
than  12  hours,  subtract  it  from  24  hours;  the  remainder  will 
be  the  longitude  East. 

Remember  that  t,  the  H.  A.  of  the  sun  at  maximum  alti- 
tude is  easterly  when  the  sun  and  zenith  are  separating,  west- 
erly when  approaching  (Art.  246) ;  and,  if  easterly,  that  t  is 


LONGITUDE  BY  EQUAL  ALTITUDES  587 

additive  to  the  mean  of  chronometer  times,  if  westerly  it  is 
subtractive  from  that  mean  to  give  the  C.  T.  of  local  appar- 
ent noon. 

In  other  words,  when  the  ship  and  sun  are  approaching, 
the  H.  A.  at  maximum  altitude  is  subtracted  from  the  mean 
of  chronometer  times  to  give  the  chronometer  time  of  merid- 
ian passage ;  when  the  ship  and  sun  are  separating,  the  reverse 
rule  holds. 

Ex.  212.— On  April  22,  1905,  latitude  by  D.  K.  26°  00'  K, 
longitude  by  D.  E.  46°  03'  W.,  observed  from  the  bridge 
of  a  vessel  steaming  315°  (true)  20  knots  per  hour  equal 
altitudes  of  sun's  lower  limb  as  follows:  In  the  forenoon 
HX  75°  45'.  W.  llh  48m  12s.  C— W  3h  Olm  28s.  Chronom- 
eter slow  of  G.  M.  T.  2m  05s.  I.  C.  +  1'.  Height  of  eye  45 
feet.  After  the  lapse  of  about  20  minutes,  the  same  limb  of 
the  sun  was  observed  at  the  same  altitude,  W.  12h  08m  14s. 
C— W  3h  Olm  27s.  The  chronometer  error,  I.  C.,  and  height 
of  eye  as  before.  Required  the  longitude  at  noon. 

The  sun's  declination  corrected  for  longitude  is  N.  12°  06' 
16",  the  H.  D.  N.  50".59,  Eq.  of  T.  corrected  for  longitude  lm 
28s  (+  to  M.  T.),  and  from  Table  26,  Bowditch,  A0/t  =  7".15. 

Course.    I  Distance.  IN     I    \V     I   L0       26°  07'  03"  N 
315°        I          20'          I  14'.1   I  14M  I   D  =  15'.71  W 

4 

Hourly  change  of  Long.  =  D  =  628.84  W  expressed  as  time. 

// 

Observer's  change  in  Lat.          14'.1   N  per  hour  or  14.1      N  per  minute 
Change  in  sun's  dec.  50".59  N  per  hour  or  0.843  N  per  minute 

Ac  (Art.  246)  combined  velocity  of  separation  =      13.257  N  per  minute 

From  formula  (203),  t=  -*~  =  ^|~  =  0-».927  =  0<°  55" .62. 

As  observer's  zenith  and  the  body  are  separating,  tf  the  H. 
A.  of  maximum  altitude  is  easterly,  and  as  the  ship  changes 
longitude  to  the  westward  at  the  rate  of  62S.84  per  hour,  or 
Os.97  in  Om.927,  the  corrected  H.  A.  is  Om  56S.59  and  the  time 


588 


NAUTICAL  ASTRONOMY 


of  maximum  altitude  is  Om  5 6s.  59  before  the  instant  of  upper 
meridian  transit. 


A.  M.  Times. 

P.  M.  Times. 

Time  of  Apparent  Noon. 

h      m      § 

Y.                 11    48    12 
:-W             3    01    28 

h      m       a 
W.                         12    08    14 
C-W                     3    01    27 

G.M.T.  of  noon  3  02  42.09 

Eq.  of  T.             +1  28 

3.  C.              +2   05 

C.  C.              +            2    05 

G.A.T.  of  noon  3  04  10.09 

i.  M.  T.        2    51    45 

P.  M.  G.  M.  T.     3    11    46 
A.  M.  G.  M   T.    2    51    45 

Long.   =    46°  02'  31".3  West 

Mid.  G.  M.  T.      3    01    45.5 
t  East                              56.59 

G.  M.  T.  of  ap- 
parent noon      3    02    42.09 

Single  and  Double  Altitudes. 

278.  What  has  been  said  about  the  general  subject  of  single 
and  double  altitudes,  their  advantages,  uses,  limiting  condi- 
tions, etc.,  under  the  head  of  chronometer  error  (Art.  268), 
applies  to  the  subject  of  longitude  when  these  methods  of  ob- 
servation are  used,  either  afloat  or  ashore. 

The  finding  of  longitude  by  these  methods  has  been  fully 
explained  in  the  chapter  on  "  Solutions  of  the  Astronomical 
Triangle"  (Arts.  226-232).  The  question  of  finding  lon- 
gitude at  sea  will  be  further  amplified  under  the  head  of 
Sumner  lines. 


CHAPTEE  XIX. 

STJMNER'S  METHOD.— SUMNER  LINES  OR  LINES  OF 
POSITION. 

279.  A  ship  approaching  the  entrance  to  Chesapeake  Bay, 
with  Cape  Charles  light  in  sight  on  the  starboard  side  and 
Cape  Henry  light  on  the  port  side,  at  a  given  moment,  may 
be  located  at  one  of  two  points  on  a  Mercator  chart,  without 
bearings  having  been  taken,  if  the  navigator  knows  the  dis- 
tance from  each  lighthouse.     Say  the  ship  is  p  miles  from 
Cape  Charles  and  q  miles  from  Cape  Henry;  with  a  pair  of 
dividers  and  a  radius  of  p  miles,  describe  a  circle  on  the  chart 
with  Cape  Charles  light  as  a  center.     Being  p  miles  distant 
from  that  lighthouse,  the  ship  is  somewhere  on  that  circle 
which  is  a  line  of  position,  passing,  as  it  does,  through  the 
position  of  the  ship.     If  a  bearing,  sounding,  or  other  deter- 
mining factor  can  be  gotten,  a  fix  may  be  obtained. 

If  a  second  circle  be  described,  with  a  radius  of  q  miles, 
from  Cape  Henry  light  as  a  center,  we  shall  have  a  second 
line  of  position,  at  some  point  of  which  also  the  ship  is 
located. 

Being  on  both  circles  at  the  same  instant,  the  ship  must  be 
at  one  of  the  two  intersecting  points.  If  the  ship's  position 
is  further  restricted  by  a  sounding,  by  latitude  or  by  longitude, 
to  the  vicinity  of  one  point,  the  other  one,  as  a  position,  is 
eliminated. 

280.  Lines  of  position  and  how  determined. — As  previously 
defined,  a  line  passing  through  a  position  of  the  ship,  whether 
a  position  by  D.  E.  or  by  observation,  is  a  line  of  position;  it 


590  NAUTICAL  ASTRONOMY 

may  be  straight  or  curved,  and  it  may  be  determined  from 
celestial  bodies  as  well  as  terrestrial  objects. 

To  Captain  Sumner,  an  American  shipmaster,  is  due  the 
credit  for  first  defining  a  ship's  position  upon  a  line,  which  he 
called  a  circle  of  equal  altitudes,  from  the  altitude  of  a 
heavenly  body  and  its  corresponding  G.  M.  T. ;  and  also  for 
determining  the  ship's  position  at  one  of  the  two  intersecting 
points  of  two  such  circles. 

281.  A    heavenly    body's    geographical    position. — Every 
heavenly  body  is  at  a  given  instant  of  time  in  the  zenith  of 
some  point  on  the  earth's  surface ;  this  point  is  the  geographi- 
cal position  of  the  body ;  for  the  sun,  it  may  be  called  the  sub- 
solar  point,  for  any  other  heavenly  body,  the  subastral  point. 

The  theory  being  the  same  for  all  bodies,  the  method,  as 
applicable  to  the  sun,  will  be  described. 

282.  The  sun's  circle  of  equal  altitudes. — The  sun  being  in 
the  zenith  of  a  given  place,  one-half  of  the  earth  will  be 
illuminated  (neglecting  refraction),  and  the  other  half  will 
be  in  darkness;  the  dividing  line,  called  the  circle  of  illumi- 
nation, will  be  everywhere  90°  from  the  subsolar  point. 

To  observers  anywhere  on  the  circle  of  illumination,  the  sun 
will  be  in  the  horizon ;  at  the  subsolar  point,  the  sun  will  be  in 
the  zenith,  and  therefore  its  altitude  will  be  90°.  If  the 
observer  is  at  any  intermediate  point  between  the  circle  of 
illumination  and  the  subsolar  point,  he  will  have  the  sun 
above  his  horizon  and  at  an  altitude  less  than  90°.  If  a 
plane  be  passed  through  this  intermediate  position,  parallel 
to  the  circle  of  illumination,  its  intersection  with  the  earth's 
surface  will  cut  out  a  small  circle,  at  every  point  of  which,  at 
the  given  instant,  the  sun  will  have  the  same  altitude.  This 
circle  is  called  a  circle  of  equal  altitudes  with  respect  to  the 
sun,  and  the  sun's  zenith  distance,  at  the  given  instant,  is  the 
same  at  all  points  of  the  circle. 


LINES  OF  POSITION"  591 

Observed  zenith  distance  as  radius  of  a  circle  of  position. — 

Since  the  distance  of  the  observer's  zenith  from  the  heavenly 
body,  in  minutes  of  arc,  is  the  same  as  the  observers  distance 
in  sea  miles  from  the  body's  geographical  position,  and  in  the 
case  of  the  sun  from  the  subsolar  point,  when  an  observer 
measures  the  altitude  of  the  sun,  the  complement  of  which 
is  its  zenith  distance,  he  actually  finds  his  distance  in  sea 
miles  from  a  known  spot  on  the  earth,  and  hence  locates  him- 
self on  a  circle  of  position  exactly  as  did  the  observer,  referred 
to  in  Art.  279,  who  found  himself  on  a  circle  of  position 
around  Cape  Charles  or  Cape  Henry. 

283.  Coordinates  of  the  geographical  position  of  a  heav- 
enly body. — The  geographical  position  of  a  heavenly  body  is 
located  like  any  terrestrial  point  by  its  latitude  and  longitude ; 
the  latitude  being  the  body's  declination,  the  longitude  the 
body's  Greenwich  hour  angle.  In  the  case  of  the  subsolar 
point,  the  latitude  equals  the  sun's  declination,  and  the 
longitude  the  Greenwich  apparent  time. 

Use  of  a  terrestrial  globe  in  connection  with  a  Simmer 
circle. — If  the  subsolar  point  be  located  on  a  terrestrial  globe 
and  a  circle,  whose  radius  equals  the  observed  zenith  distance 
of  the  sun,  is  drawn  on  the  globe,  with  the  subsolar  point  as 
a  center,  the  observer  will  be  somewhere  on  the  circumference 
of  this  circle;  since  the  subsolar  point  bears  in  a  given  direc- 
tion from  him,  his  position  is  in  the  opposite  direction  from 
the  subsolar  point,  so  that  the  sun's  azimuth  at  the  time  of 
the  observation  indicates  the  part  of  the  circle  on  which  the 
observer  is  situated;  and  his  position  would  be  fixed,  if, 
having  the  above  data,  he  should  find  his  latitude  or  longitude, 
or  a  second  circle  of  equal  altitudes,  projected  from  observa- 
tions of  a  second  heavenly  body,  or  from  observations  again  of 
the  sun  after  a  lapse  of  sufficient  interval  of  time,  the  obser- 
ver's position  remaining  unchanged. 

However,  this  graphic  method  cannot  be  used  for  the  reason 


592  NAUTICAL  ASTRONOMY 

that  it  is  impracticable  to  carry  a  globe  of  such,  dimensions 
as  to  admit  of  accurate  results. 

284,  On   a   Mercator   chart. — The   circles   of   equal   alti- 
tude will  appear  on  this  chart  as  shown  in  Plate  XV,  end  of 
book,  being  drawn  out  towards  the  North  and  South  points 
for  reasons  apparent  to  anyone  familiar  with  the  theory  of  the 
Mercator  projection.     These   curves   are  called   "  Curves  of 
equal  altitudes."     Fig.  121,  right-hand  side,  shows  the  curves 
at  intervals  of  10  °,  in  which  $±  is  the  geographical  position  of 
the  body  observed ;  all  these  curves  belong  to  the  same  system 
which  Sumner  called  a  system  of  illumination.     It  will  be 
noticed  that  all  these  curves  cut  the  parallels  of  latitude  and 
meridians  of  longitude  at  different  angles.     Near  the  North 
and  South  points,  the  curves  run  about  East  and  West  with 
the  parallels  of  latitude,  and  a  large  error  in  longitude  makes 
but  a  slight  error  in  latitude ;  near  the  East  and  West  points, 
the  curves  run  with  the  meridians  and  a  large  error  in  lati- 
tude makes  but  a  slight  error  in  the  resulting  longitude;  at 
intermediate  points,  the  curves  cut  the  parallels  and  meridians 
at  varying  angles,  so  that  the  error  in  longitude  due  to  a  given 
error  in  latitude  depends  on  the  body's  azimuth. 

These  facts  can  be  regarded  as  additional  proofs  that  bodies 
should  be  observed  for  latitude  when  on  or  near  the  meridian, 
and  for  longitude  when  on  or  near  the  prime  vertical. 

Determination  of  points  on  the  curve. — If  an  observer  has 
a  given  altitude,  different  assumed  latitudes,  within  the  limits 
of  the  curve,  will  give  him  different  longitudes,  and  vice 
versa.  Each  latitude  will  give  two  points,  one  for  an  altitude 
East,  one  for  an  altitude  West  of  the  meridian  through  the 
observed  body.  By  assuming  a  sufficient  number  of  coordi- 
nates, the  whole  curve  may  be  plotted. 

285.  (a)   Double  altitude  observations. — Suppose  that  on 
April  16,  1905,  p.  m.  time,  an  observer  at  sea  on  the  North 


LINES  OF  POSITION 


593 


Atlantic  Ocean,  observes  the  true  altitude  of  the  sun's  center 
to  be  50°,  the  G.  M.  T.  of  observation  being  lh  15m  30s. 

Taking  the   required   data   from   the  Nautical   Almanac 
(1905)  for  the  given  G.  M.  T.,  the  sun's  declination  is  found 

I30°12001100100°900  80°  70°  60°  50°  40°30°  20°  10°  0°  10°  20°  30°  40°  50° 


W 

es,L 

ong. 

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East 
\ 

Lor 

g. 

60° 

50° 
40° 

30° 
20° 
10° 
0° 
10° 
20° 
30° 

40° 
50° 

60° 
70° 

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/ 

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FIG.  121. 

to  be  10°  N.,  the  equation  of  time  Om  06s,  additive  to  mean 
time,  and  the  G.  A.  T.  lh  15m  36s.  Hence  the  latitude  of  the 
subsolar  point  8±  is  10°  K,  and  its  longitude  lh  15m  36s  West, 
or  18°  54'  W. 

From  this  point  8±  as  a  center,  -the  curve  of  equal  altitudes 
witn  radius  of  40°  will  be  the  right-hand  curve  TVXS'  on  the 


594  NAUTICAL  ASTRONOMY 

Mercator  chart  (Fig.  121).  This  curve  tells  us  nothing  more 
than  the  bare  fact  that  the  observer  is  somewhere  on  its  cir- 
cumference. However,  if  the  bearing  of  the  sun  at  the  instant 
of  sight  is  given,  the  quadrant  containing  the  position  is 
indicated.  If,  in  example,  the  sun  bears  southward  and 
eastward,  the  ship  is  in  the  N~W.  quadrant.  Having  obtained 
the  curve  and  bearing  of  the  sun,  if  either  latitude  or  longi- 
tude is  given,  the  ship's  position  is  determined. 

Again,  suppose  that  after  the  lapse  of  3h  24m  24s  a  second 
observation  shows  the  sun's  altitude  to  be  40°.  During  this 
interval,  the  sun  in  its  diurnal  path  will  have  passed  to  the 
westward  at  the  rate  of  15°  of  longitude  per  hour,  carrying 
with  it  its  geographical  position  and  its  system  of  curves  of 
equal  altitudes.  The  G-.  A.  T.  becomes  4h  40m,  or  the  longi- 
tude of  the  subsolar  point  S2  is  70°.  The  declination  is 
K  10°  03'  01",  or  the  latitude  of  S2  is  K  10°  03'  01",  the 
zenith  distance  is  50°,  the  curve  on  the  Mercator  chart  is 
N'XS'9  and  the  ship  is  somewhere  on  this  curve.  What  we 
know  now  is  that  at  the  first  observation  the  ship  was  on  the 
right-hand  curve  NXS,  and  at  the  second  observation  she  was 
on  the  left-hand  curve  N'X8'9  therefore,  if  the  ship  did  not 
change  her  position  in  the  interval  between  the  observations, 
she  was  at  one  or  the  other  of  the  two  points  (X  or  Y)9  in 
which  the  curves  intersected,  and  the  one  which  was  the 
observer's  position  depends  on  the  sun's  bearings  at  the  times 
of  observation. 

(b)  In  case  the  observer  changes  his  position  between  the 
observations. — If  the  observer,  whose  position  is  somewhere 
on  the  right-hand  curve,  can  be  supposed  to  make  an  instan- 
taneous change  of  position,  through  a  distance  of  N  sea  miles 
directly  towards,  or  directly  away  from  the  geographical 
position  S19  on  a  great  circle  passing  through  S19  the  sun's 
altitude  will  be  increased  or  diminished  by  N  minutes  of  arc ; 
if  the  course  is  kept  at  right  angles  to  the  bearing  of  the  sun, 


LINES  OF  POSITION  595 

he  will  keep  on  his  original  curve  of  altitude ;  if  the  course  is 
at  intermediate  angles,  the  altitude  will  be  changed  propor- 
tionally (the  change  being  expressed  in  Art.  213  by  for- 
mula Aft  —  d  cos  (C~Z)),  and  the  observer  will  be  on 
another  circle  of  equal  altitudes,  belonging,  however,  to  the 
same  system  of  circles,  that  is,  the  system  having  8^  as  a 
center. 

In  going  then  from  a  point  on  one  circle  of  equal  altitudes, 
which  was  the  ship's  position  at  the  first  altitude,  on  a  certain 
course  and  distance  made  good,  the  observer  arrives  at  a  point 
on  another  circle  of  equal  altitudes,  of  the  same  system,  how- 
ever. The  altitude  at  this  latter  circle  corresponds  to  what 
would  have  been  observed  there  at  the  instant  of  the  first 
observation  on  the  original  circle. 

The  circle  of  position  can  then  be  found  after  a  run  to  the 
place  of  a  second  observation,  either  by  reducing  the  first 
altitude  to  what  it  would  have  been  at  the  place  of  the  second 
observation,  at  the  time  of  the  first  observation  (see  Art. 
213)  ;  or  by  taking  a  point  in  the  first  curve,  representing  the 
ship's  approximate  position,  laying  off  the  course  and  distance 
made  good  in  the  interval  from  it  to  a  second  point,  and  then 
drawing  through  this  second  point  a  curve  parallel  to  the  cor- 
responding part  of  the  first  curve  of  altitude.  The  intersection 
of  this  transferred  curve  with  the  curve  of  altitude  of  the 
second  observation  will  give  the  ship's  position  at  the  time  of 
the  second  observation.  The  general  method  described  in  the 
two  sections  of  this  article  is  known  in  modern  navigation  as 
"  Sumner's  double  altitude  method." 

Ordinarily,  it  is  impracticable  to  plot  circles  of  position  on 
a  Mercator  chart  without  previous  calculations  for  many 
apparent  reasons,  but  under  certain  circumstances  this  may 
be  done,  and  from  two  circles  the  fix  may  be  found  with  con- 
siderable accuracy. 

In  certain  cases,  as  may  happen  in  the  tropics,  the  sun  may 


596  NAUTICAL  ASTRONOMY 

be  observed  when  close  to  the  zenith,  say  within  a  degree  or 
so;  the  subsolar  point  located  by  its  latitude  and  longitude 
(the  latitude  being  the  sun's  declination  and  longitude  the 
G-.  A.  T.  of  observation) ;  and  that  portion  of  the  circle  near 
the  ship's  D.  E.  position  drawn  with  the  true  zenith  distance 
as  a  radius. 

After  the  sun's  azimuth  has  altered  from  25°  to  30°,  and 
under  such  circumstances  it  will  do  so  in  a  very  short  time, 
draw  a  second  arc  as  the  result  of  a  second  observation. 
Transfer  the  first  arc  for  the  run  between  sights,  and  the 
intersection  of  the  transferred  arc  with  the  arc  corresponding 
to  the  second  observation  will  give  the  fix  With  a  fair  degree 
of  accuracy,  and  this  without  any  of  the  usual  calculations. 

In  using  the  method  of  double  altitudes,  there  should  be  a 
change  of  bearing  of  the  sun  between  observations  of  at  least 
two  points;  of  course,  the  nearer  the  change  is  to  90°,  the 
more  nearly  the  resulting  lines  of  position  run  at  right  angles 
to  each,  other,  and  the  better  the  cut. 

Rapidity  of  change  of  azimuth  dependent  on  L  and  d. — 
The  rapidity  of  change  of  the  sun's  azimuth  will  depend  on 
the  values  of  L  and  d,  and  the  time  an  observer  has  to  wait 
for  that  body  to  undergo  a  desired  change  of  bearing  may  be 
found  by  inspection  of  the  azimuth  tables.  The  greater  the 
difference  between  the  values  of  L  and  d,  the  smaller  will  be 
the  elapsed  interval  for  a  given  change;  for  this  reason,  the 
interval  in  winter  months  will  be  smaller  for  observations  of 
the  sun. 

When  the  latitude  and  declination  are  nearly  the  same,  it 
will  be  impossible  to  work  the  double  altitude  problem  from 
observations  on  the  same  side  of  the  meridian;  however, 
having  obtained  an  a.  m.  sight,  if  the  meridian  observation 
is  lost,  it  will  not  take  long  for  the  sun  after  crossing  the 
meridian  to  alter  the  first  azimuth  90°,  so  that  lines  giving 


LINES  OF  POSITION  597 

excellent  cuts  may  be  obtained  by  combining  observations  on 
both  sides  of  the  meridian. 

286.  Simultaneous    observations. — The   principles    of   the 
Sumner  double  altitude  method,  as  explained  in  the  case  of 
the  sun  in  the  preceding  article,  apply  as  well  to  any  other 
heavenly  body;  but,  as  a  general  thing,  when  one  star  may  be 
observed,  others  are  available,  so  that  two  (or  more)  may  be 
observed    at   one    time    and    the    ship    may    be    located    at 
one  of  the  two  intersecting  points  of  the  resulting  circles 
of  position.     Suppose  two  stars  are  observed  at  the  same  mo- 
ment at  a  given  place;  that  one,  whose  subastral  point  is  $± 
(Fig.  121),  has  an  altitude  of  50°  and  bears  southward  and 
eastward,  and  that  the  other,  whose  subastral  point  is  S2 , 
has  an  altitude  of  40°  and  bears  southward  and  westward. 
The  result  of  the  first  observation  locates  the  ship  on  the  NW. 
arc  of  the  circle  of  position  NXS,  the  result  of  the  second  on 
the  NE.  arc  of  the  circle  N'XS',  and,  therefore,  at  their 
northern  intersection  X.     Such  observations  are  known  as 
simultaneous    observations;    for    these    observations,    bodies 
should  be  so  selected  that  the  resulting  circles  or  lines  of 
position  will  cut  at  good  angles,  not  less  than  30°. 

Advantages  of  simultaneous  over  double  altitude  observa- 
tions.— The  former  are  preferred  to  the  latter  as  the  position 
of  the  ship  may  be  obtained  at  once  without  an  interval  of 
waiting  and  the  errors  of  the  run  when  there  is  a  change  of 
position;  besides,  a  third  line  may  be  obtained  and  the  fix 
from  two  lines  either  verified  or  disproved,  the  fix  being  veri- 
fied when  the  three  lines  have  practically  the  same  point  of 
intersection. 

It  is  probable  that  the  navigator  may  have  such  a  number 
of  first  or  second  magnitude  stars  to  select  from  that  good 
observations  may  be  obtained  in  all  latitudes. 

287.  Relation  between  circles  of  equal  altitude  and  the 
astronomical  triangle. — Let  Fig.  122  represent  a  projection 


598 


NAUTICAL  ASTRONOMY 


on  the  horizon  of  a  point  8  the  geographical  position  of  a 
heavenly  body;  PQ,  the  meridian  of  that  point;  PG,  the 
meridian  of  Greenwich ;  PZ±  and  PZ2,  the  meridians  of  places 
on  the  earth's  surface  having  at  the  same  instant  of  time  the 
same  altitude  of  the  body  8.  Z-LZ2Z3  is  a  Sumner  curve 
or  a  circle  of  equal  altitudes  with  respect  to  the  body  8;  QS, 
the  latitude  of  the  geographical  position  equals  the  body's  de- 
clination; GPS,  the  Greenwich  hour  angle  of  the  body,  is  the 


FIG.  122. 


longitude  of  8.  The  triangles  ZfS  and  Z2P8  are  projections 
of  astronomical  triangles.  The  angle  ZfS  is  the  hour  angle 
-ofjhe  body  at  a  place  Z±  on  the  circle  of  equal  altitudes; 
ZfS,  the  hour  angle  of  the  same  body  at  the  same  instant  at 
the  place  Z2 ,  also  on  the  same  circle. 

Since  the  Greenwich  hour  angle,  the  declination,  and  the 
altitude  are  the  same  at  Zi9Z2,  and  other  places  on  the  same 
circle  of  altitude,  the  astronomical  triangles  Z^PS,  Z2P8,  etc., 
have  two  sides  of  one  equal  to  two  sides  of  the  others,  one 
side  equal  to  90°— h  and  the  other  equal  to  90°—^  but  they 
differ  in  the  values  of  the  third  side  (PZ± ,  PZ2 ,  etc.),  which 
side  is  the  complement  of  the  latitude,  and  also  in  the  values  of 


LINES  OP  POSITION-  599 

the  local  hour  angles,  Z^PS,  Z2PS,  etc.  The  hour  angles  of 
the  body  at  the  local  meridian,  and  hence  the  longitudes,  are 
dependent  on  the  different  assumed  values  of  the  latitude 
when  solving  the  astronomical  triangle  with  given  values  of 
h  and  G.  M.  T.  For  the  sun,  the  hour  angles  ZJ?8,  Z2PSf  etc., 
are  the  local  apparent  times,  or  24  hours — rthose  apparent 
times,  according  as  the  times  are  less  or  greater  than  12  hours'; 
GPS  is  the  Greenwich  apparent  time,  or  24  hours — that  time, 
according  as  the  apparent  time  at  Greenwich  is  less  or  greater 
than  12  hours ;  and  GPZ2  and  GPZ^  are  the  longitudes  from 
Greenwich,  respectively,  of  Z2  and  Z±. 

By  assuming  latitudes  and  finding  the  corresponding  longi- 
tudes, or  by  assuming  longitudes  and  finding  the  correspond- 
ing latitudes,  any  number  of  points  of  the  curve  may  be  found 
and  the  whole  circle  projected. 

288.  Kule  for  assuming  coordinates. — The  rule  for  assum- 
ing coordinates,  based  on  what  has  been  said  as  to  the  varying 
angles  at  which  the  curve  of  equal  altitudes  cuts  meridians 
and  parallels  of  latitude    (Art.   284),   and  the   demonstra- 
tions (see  Arts.  237  and  248)  as  to  the  best  times  to  observe 
for  latitude  or  longitude,  is  as  follows:  assume  latitudes  and 
solve  for  longitudes  when  the  body's  ZN  lies  between  45°  and 
135°,  or  225°   and  315°;  otherwise,  assume  longitudes  and 
solve  for  latitudes. 

289.  Actual  sea  practice  and  method  of  determining  the 
line, — In  actual  practice  at  sea,  it  is  never  necessary  to  de- 
termine more  than  a  small  portion  of  the  circle  of  equal 
altitudes,  since  the  observer's  position  is  generally  known,  to 
be  within  certain  limits,  both  of  latitude  and  of  longitude. 
This  small  portion  is  the  only  part  to  be  considered;  it  is 
called  a  line  of  position,  and  is  at  right  angles  to  the  heavenly 
body's  true  bearing.     If  three  or  more  coordinates  are  as- 
sumed, especially  if  they  are  far  apart  or  the  body's  altitude 
is  great,  the  line  may  be  a  curved  line.     It  is  customary, 


600  NAUTICAL  ASTRONOMY 

however.,  when  working  for  longitudes,  to  assume  two  lati- 
tudes differing  say  ~by  20' ,  or  when  working  for  latitudes  to 
assume  two  longitudes  differing  by  two  minutes  of  time  or 
30'  of  arc;  in  loth  cases,  the  dead  reckoning  position  should 
be  between  the  assumed  coordinates.  For  such  short  dis- 
tances the  chord  thus  obtained  is  practically  equal  to  the 
included  arc  of  the  circle.  This  is  known  as  the  "  method  of 
chords  "  and  for  years  has  heen  the  practice  of  the  officers  of 
the  U.  S.  Navy.  It  is  evident  that,  in  the  case  of  a  line  thus 
determined,  its  angle  with  the  meridian  may  be  found  by 
either  middle  latitude  or  Mercator  sailing,  and  thence  the 
true  azimuth  of  the  body  whose  bearing  is  at  right  angles  to  it. 

Since  the  circle  of  equal  altitudes  is  at  right  angles  to  the 
true  bearing  of  the  body,  a  tangent  to  the  circle  at  a  given 
point,  and  for  short  distances  either  side,  may  be  taken  as 
practically  equal  to  the  arc  between  the  same  limits.  There- 
fore, to  determine  a  line,  assume  a  latitude  and  find  the  corre- 
sponding longitude,  or  assume  a  longitude  and  find  the  corre- 
sponding latitude,  both  assumptions  within  the  limits  of  the 
curve,  thus  determining  one  point  of  the  circle  of  equal 
altitudes. 

The  true  azimuth  of  the  body  for  the  instant  of  observation 
having  been  determined  in  one  of  three  ways  (1)  from  the 
azimuth  tables  (Art.  221)  or  an  azimuth  diagram;  (2)  by 
observation,  the  compass  bearing  being  corrected  for  variation 
and  deviation  of  the  compass;  (3)  by  solution  of  the  astro- 
nomical triangle  (Arts.  218  and  219);  (2)  and  (3)  not 
being,  however,  the  usual  practice  at  sea;  a  line  is  drawn 
through  the  determined  point  at  right  angles  to  the  body's 
true  bearing.  This  line  is  a  line  of  position,  and  this  method 
of  determining  it  is  known  as  the  "  method  of  tangents." 

290.  To  define  a  line  of  position. — From  what  has  been 
said  in  previous  articles,  a  line  may  be  defined  in  one  of  two 


LINES  OF  POSITION  601 

ways — when  determined  by  the  chord  method,  it  is  defined  by 
its  two  points  A19  A2,  thus : 

18°  50'  S.  .    (  19°  10'  S. 

2°  46'  24"  W.  2       2°  37'  31"  W. 


j.  t'O     v  »T  \s     k/v/j.AJ.1. 


When  determined"  loj  the  tangent  method,  it  is  defined  loj  its 
one  position  point  A  and  its  direction  thus : 

f!9°  00'  S  fAzimuth  of  body  ZN=67°10' 

Position  Pt.  A  \  no  ±-\t  Ke/fw"  J  obtained  from  azimuth  tables, 

I  A  L  58  W.  •<  giyen  L= igoQ^  £r^8h01m528, 

Line  of  Position  337°  10'  I  d  =  N  11°  50'. 

From  the  data  of  line  A^A^  by  middle  latitude  sailing, 
the  direction  of  the  line  is  found  to  be  33 7°. 2  and  the  body's 
true  bearing,  ZN,  67°. 2.  In  other  words,  the  line  has  the 
same  direction,  however  determined. 

Coordinates  "computed"  and  "by  observation." — In  this 
work,  wherever  they  occur,  the  terms  Lat.  and  Long,  by 
observation  will  be  taken  as  applying  only  to  the  ship's  posi- 
tion or  fix;  the  term  computed  latitude,  as  referring  to  that 
obtained  from  a  sight  by  using  the  D.  E.  longitude  or  an 
assumed  longitude;  and  the  term  computed  longitude,  as  re- 
ferring to  that  obtained  by  using  a  D.  E.  latitude  or  an 
assumed  latitude. 

The  method  of  determining  the  line  and  the  methods  of 
finding  the  intersection  of  two  lines  will  be  considered  in 
Arts.  295-310. 

291.  Uses  of  a  Sumner  line.— If  the  G.  M.  T.  and  altitude 
of  the  heavenly  body  are  correct,  the  line  determined  from  the 
data  passes  through  the  position  of  the  ship,  and  if  the  line, 
or  the  line  produced,  passes  through  a  lighthouse,  point  of 
land,  or  a  danger,  the  direction  of  the  line  gives  at  once  the 
bearing  of  that  particular  object;  if  desiring  to  make  that 
light  or  point  of  land,  the  navigator  knows  the  course  to  steer ; 
to  avoid  the  object,  if  a  danger,  it  is  only  necessary  to  run  at 


602 


NAUTICAL  ASTRONOMY 


right  angles  to  the  direction  of  the  line  for  a  safe  distance, 
and  then,,  by  changing  the  course  not  more  than  90°  from 
this  last  course,  the  ship  will  go  clear. 

If  in  Fig.  123,  A^A^  is  a  line  of  position  passing  through 


FIG.  123. 


FIG.  124. 


a  lighthouse  or  point  of  land  B,  the  course  to  be  steered  to 
make  B  is  the  direction  of  the  line  towards  B. 

If  B  is  a  danger,  run  the  course  and  distance  represented 
by  cd,  or,  if  safe,  run  a  proper  distance  in  the  direction  ce; 
then  the  direction  A2A2  will  clear  the  danger. 

If  the  direction  of  the  line  is  into  the  port  of  destination, 
the  course  in  will  be  known;  if  its  direction  is  towards  a 
point  to  one  side  of  the  entrance,  A1A1  (Fig.  124),  draw  a 
line  A2A2  on  the  chart  from  the  entrance  EF,  parallel  to  the 
line  of  position,  shape  a  course  cd  at  right  angles  to  A^A± , 
or,  if  safe,  a  course  in  direction  ce  till  the  vessel  arrives 
on  the  parallel  line  A2A2-  then  steer  in  its  direction  for 


CHAKT  INTERSECTIONS  603 

the  entrance.  If  the  line  runs  parallel  to  the  coast  (BE, 
Fig.  124),  the  distance  off  shore  will  be  known. 

A  fix  may  be  obtained  by  a  verified  sounding  or  by  a  bear- 
ing of  an  object  of  known  position  on  the  chart;  and  in  this 
connection,  when  in  the  vicinity  of  dangers,  attention  is  again 
called  to  the  fact  that,  if  a  line  is  obtained  from  an  observa- 
tion of  a  body  on  the  prime  vertical,  the  longitude  will  be  well 
determined;  if  from  an  observation  of  a  body  on  the  meridian, 
the  latitude  will  be  well  determined;  even  though  the  other 
coordinate  may  be  somewhat  in  error. 

Owing  to  the  fact  that  a  line  is  always  at  right  angles  to  the 
bearing  of  the  body,  it  is  often  possible,  especially  at  night 
when  heavenly  bodies  in  all  directions  are  available,  to  get  a 
line  running  in  any  desired  direction,  so  as  to  show  the  bearing 
of  land,  distance  of  coast,  etc. 

If  an  observation  of  a  heavenly  body  is  taken  when  bearing 
directly  abeam* — and  the  opportunities  are  many  for  so  observ- 
ing not  only  stars  but  the  sun — the  resulting  line  of  position 
will  be  in  the  direction  of  the  course;  and  if  the  line  leads 
clear  of  danger  the  navigator  may  keep  his  course,  if  towards 
danger  he  may  run  off  90°  for  a  safe  distance,  then  resume 
his  course,  clearing  the  danger. 

Two  lines  intersecting  at  angles  of  not  less  than  30°  (90° 
preferred)  will  give  good  fixes.  When  possible,  it  is  better  to 
verify  this  fix  by  a  third  line. 

During  morning  or  evening  twilight,  or  moonlight,  when 
stars  are  visible,  several  may  be  observed  at  the  same  time, 
and  they  may  be  so  selected  that  the  corresponding  lines  will 
cut  at  excellent  angles,  and  hence  give  excellent  fixes. 

GRAPHIC  OR  CHART  INTERSECTIONS. 

292.  Finding  the  noon  position  on  a  Mercator  chart  and 
the  intersection  of  a  line  with  another  moved  parallel  to 
itself  for  the  run  between  observations. — The  parallel  of  the 


604 


NAUTICAL  ASTRONOMY 


latitude  found  at  noon  is  nothing  more  nor  less  than  a  line 
of  position  obtained  when  the  body  is  observed  on  the 
meridian,  and  the  noon  position  is  the  intersection  of  this 


44°N 


FIG.  125. 


parallel  with  the  a.  m.  line  moved  for  the  run  between  obser- 
vations 

Let  Fig.  125  be  a  section  of  a  Mercator  chart  with  Sum- 
ner  lines  plotted  thereon.  Suppose  the  a.  m.  observation 
gave  a  line  A^A^  (Fig.  125),  and  that  the  run  to  noon  was 
120°  (true)  15  miles  (in  the  direction  of  and  equal  to  cd), 
and  the  noon  latitude  was  that  of  the  parallel  MM;  then  the 
intersection  of  MM  with  A2A2,  the  forenoon  line  moved  up 
for  the  run,  was  the  noon  position. 


UNCERTAINTY  IN  DATA  605 

Again,  in  Fig.  125  c^  is  a  line  of  position  obtained 
by  working  a  <f>"<j>'  sight  of  the  sun  for  latitude,  having  as- 
sumed longitudes  8°  30'  W.  and  9°  W.  After  sailing  211°  true 
21.4  miles  (c^d)  and  205°  10.8  miles  (dc2),  a  second  line 
did-L  was  found  from  an  observation  of  the  sun  worked  as  a 
time  sight;  it  is  required  to  find  the  ship's  position  at  the 
second  observation.  Having  plotted  on  the  Mercator  chart 
the  first  line  c^  by  its  coordinates,  it  is  apparent  that  the 
ship  is  somewhere  on  the  line  at  the  time  of  the  observation, 
though  the  exact  point  is  unknown.  From  any  point  of  this 
line  CjC-u  lay  off  the  true  courses  and  distances  run  (in  this 
case  in  the  directions  c^d  and  dc2)  and  through  the  de- 
termined point  draw  c2c2  parallel  to  c^;  the  ship  at  the 
time  of  the  second  observation  is  on  the  line  C2c2 .  The 
second  line  by  observation  is  plotted  by  its  coordinates  and 
intersects  c2c2  in  y  which  is  the  position  of  the  ship  at  the 
time  of  the  second  observation. 

The  data  for  the  two  lines  and  for  the  ship's  run  of  the 
second  case  in  this  article  are  given  below. 
First  line  C&          A'±     8°  30'          W  )  A/2     9°  00'          W) 
L\  45°  04'  03"  N  j  L'2  45°  22'  12"  N  J 

Second  line  d&    L\  44°  26'  40"  TX\L"2  44°  46'  40"  N"  ) 
A"±     8°  47'  44"  WJ  A"2     8°  52'  26"  W  J 

Run  between  lines 

211°  (true)  21.4  miles. 
205°  (true)  10.8  miles. 

293.  Uncertainty  in  G.  M.  T. — If  there  is  an  uncertainty 
in  the  Gr.  M.  T.,  parallels  may  be  drawn  on  either  side  of  the 
line,  at  a  distance  in  longitude  equal  to  the  amount  of  the 
uncertainty,  so  that  the  true  position  will  then  be  restricted 
in  the  case  of  a  single  line  to  a  belt  instead  of  a  line,  and  in 
case  of  two  lines  to  the  area  of  a  small  parallelogram. 

294.  Uncertainty  in  altitude. — If  there  is  an  uncertainty 


60G 


NAUTICAL  ASTRONOMY 


in  altitude,  parallels  to  the  line  may  be  drawn  each  side,  at  a 
perpendicular  distance  from  it  in  nautical  miles  equal  to  the 
number  of  minutes  of  error  in  altitude. 

For  a  given  uncertainty,  to  illustrate  say  V  of  altitude, 
when  one  body  is  on  the  meridian  and  the  other  on  the  prime 
vertical,  the  position  may  be  anywhere  in  a  square,  with  a 
maximum  uncertainty  of  2'  both  as  to  latitude  and  longitude. 
Thus  if  aa'  and  W  (Fig.  126),  represent  the  two  lines  of  posi- 
tion, the  observer  would  be  at  0,  provided  there  was  no  error ; 
but  to  allow  for  a  possible  error  of  V  of  altitude,  +  or  — , 
lines  must  be  drawn  on  each  side  at  the  perpendicular  distance 
of  one  sea  mile.  The  observer  may  be  at  1,  2,  3,  or  4,  or  any- 
where within  the  square,  making  the  limits  of  uncertainty  two 
miles  for  both  latitude  and  longitude. 


FIG.  126. 

For  a  difference  of  azimuth  of  eight  points,  where  neither 
body  is  on  the  meridian  or  prime  vertical,  the  rectangle  will 
be  shifted,  and  the  uncertainty  in  latitude  and  longitude  will 
increase  till  when  the  lines  run  NE.  and  NW.  (SW.  and  SE.), 
each  becomes  a  maximum;  the  position  may  vary  in  latitude 
2.8  sea  miles,  and  in  longitude  the  same  amount  (Fig.  127). 

For  a  difference  of  azimuth  greater  or  less  than  eight 
points. — When  the  difference  of  azimuth  of  the  two  lines  is 


INTERSECTION  OF  LINES  607 

greater  or  less  than  90°,  an  error  in  altitude,  +  or  ( — ),  will 
affect  the  possible  position  of  the  ship  so  as  to  make  the  un- 
certainty in  latitude  greater  and  in  longitude  less,  or  vice 
versa,  according  to  the  direction  in  which  the  parallelo- 
gram is  elongated.  For  instance,  if  the  difference  of  azi- 
muth is  small  and  the  mean  azimuth  is  near  East  or  West, 
as  in  Fig.  128,  where  aa'  and  W  are  the  lines  of  position,  it 
is  seen  that  the  possible  variation  in  longitude  (1  to  3)  is 


FIG.  127.  FIG.  128. 

small  compared  to  the  variation  in  latitude  (2  to  4).  So 
that,  whilst  longitude  may  be  determined  when  the  mean 
azimuth  is  near  E.  or  W.,  the  latitude  may  be  far  out  when 
the  difference  of  azimuths  is  small.  Exactly  the  reverse 
is  true  when  the  mean  azimuth  is  near  N.  or  S.  and  the  differ- 
ence of  azimuth  is  small. 

Finding  the  intersection  of  Sumner  lines. 
295.  The  intersection  of  lines  of  position  may  be  found : 

(1)  By  plotting,  on  a  Mercator  chart  of  the  locality  in 
which  the  ship  may  be,  the  lines  determined  (a)  by  the  chord 
method,  (b)  by  the  tangent  method. 

(2)  By  computation. 


608  NAUTICAL  ASTRONOMY 

296.  (a)  The  plotting  of  lines  determined  by  the  chord 
method. — If  a  line  of  position  is  determined  from  celestial 
observations  by  assuming  two  latitudes  and  finding  the  corre- 
sponding longitudes,  or,  by  assuming  two  longitudes  and  find- 
ing  the    corresponding   latitudes,   the    assumed   coordinates 
being  about  equally  distant  each  side  of  the  dead  reckoning 
position,  it  is  plotted  on  a  Mercator  chart  by  locating  the  two 
points  thus  determined  and  drawing  a  straight  line  between 
them. 

If  a  second  line  is  plotted  in  the  same  way,  the  observer 
will  be  at  the  intersection  of  this  second  line  with  the  first, 
provided  there  has  been  no  change  of  his  position  in  the 
interval  between  observations;  however,  if  there  has  been  a 
change,  then  the  observer's  position  will  be  at  the  intersection 
of  the  second  line  with  the  first  line  after  having  been  moved 
parallel  to  itself  for  the  run  in  the  interval.  The  principles 
involved  are  shown  in  Fig.  125. 

In  case  the  observation  is  of  a  body  on  the  meridian,  the 
line  of  position  becomes  a  parallel  of  latitude;  if  the  body  is 
observed  on  the  prime  vertical,  the  line  will  be  a  meridian. 

297.  (b)  The  plotting  of  lines  determined  by  the  tangent 
method. — In  this  method,  take  the  D.  R.  latitude  and  deter- 
mine from  the  given  observations,  by  solution  of  the  astro- 
nomical   triangle,    the    corresponding    longitude,    calling    it 
computed  longitude ;  or  take  the  D.  R.  longitude  and  find  the 
corresponding  latitude,  'Calling  it   computed   latitude.     The 
point  thus  determined  will  be  one  point  of  the  line  and  is 
plotted  on  the  chart.     Through  this  point  draw  a  line  at  right 
angles  to  the  bearing  of  the  body  for  the  instant  of  observa- 
tion, this  bearing  being  found  from  the  azimuth  tables  having 
given  L,  d,  and  t,  or  from  an  azimuth  diagram.     The  line 
drawn  will  be  a  line  of  position.     Considering  the  azimuth  of 
the  body  less  than  90 °,  the  direction  of  the  line  is  easily 
obtained  from  the  true  bearing  of  the  body  by  reversing  either 


LONGITUDE  FACTOR 


609 


letter  of  the  bearing  and  talcing  the  complement  of  the  angle. 
Thus,  if  the  body  bore  8.  30°  E.,  the  corresponding  line  of 
position  runs  N.  60°  E.  or  8.  60*  W.  * 

The  second  line  having  been  plotted  in  the  same  way,  the 
observer's  position  will  be  at  the  intersection  of  the  two  lines, 
as  explained  in  Art.  292. 

298.  Before  explaining  the  methods  of  finding  the  inter- 
section of  Sumner  lines  by  computation,  it  is  desirable  to  give 
a  few  definitions. 

Definition  of  longitude  factor. — The  longitude  factor  of  a 
line  of  position,  represent- 
ed by  the  letter  F,  is  the    LT 
change  in  lorigitude  due  to 
1'  change  in  latitude.     In 
the  case  of  a  line  determin- 
ed by  the  chord  method,  it 
is  found  directly  by  divid-    L 
ing    the    difference    of    the 
longitudes  of  the  two  points 
by  the  difference  of  their  corresponding  latitudes,  or  F  = 

j*  ~^v  =  -^ ,  where  AL  is  a  change  in  latitude  due  to  a 
change  of  AA  in  longitude,  and  vice  versa  (Fig.  129). 

Definition  of  latitude  factor. — The  latitude  factor  of  a 
line  of  position,  represented  by  /,  is  i  9  or  the  change  in 
latitude  due  to  a  change  of  1'  of  longitude.  In  this  method 
of  defining  a  line,  f=-^~^1  = — ^(Fig.  129). 

When  a  line  is  determined  by  the  tangent  method,  F  equals 
4^  and  /  equals  A^ ,  as  before ;  but  in  this  case  it  is  neces- 

A.L  A  A 

sary  to  investigate  and  ascertain  the  relation  between  AL  and 
AA  and  the  determining  quantities  of  a  tangent  line,  namely 
latitude  and  the  body's  azimuth. 

*  If  the  true  bearing  of  the  body  is  given  in  the  form  of  ZTH,  simply  add  or  sub- 
tract 90°  to  obtain  the  true  direction  of  the  line  as  estimated  from  0"  at  North 
around  to  the  right. 


FIG.  129. 


610 


NAUTICAL  ASTRONOMY 


Values  of  F  and  /  and  where  they  may  be  found. — By  ref- 
erence to  Art.  237,  it  is  seen  that,  by  differentiation  of  the 

general  equation  of  the  astronomical  triangle,  -ry  =  sec  L  cot  Z, 

dt  being  the  change  in  longitude  due  to  a  change  of  dL  in 
latitude.  Now,  if  dL  represents  one  minute  of  latitude,  dt 
=  sec  L  cot  Z  is  the  change  in  longitude  due  to  a  change  of 
1'  of  latitude,  and,  therefore,  the  longitude  factor 

F  —  sec  L  cot  Z  (236). 

This  may  be  shown  graphically  from  Fig.  130  and  Fig.  131, 
in  which  the  azimuth  is  considered  as  less  than  90°.    In  Fig. 


FIG.  130. 


FIG.  131. 


130,  the  body  bears  either  in  the  NE.  or  SW.  quadrants  and 
the  direction  of  the  line  of  position  is  NW.  or  SE.  In  Fig.  131, 
the  sun  bears  in  the  NW.  or  SE.  quadrants  and  the  direction 
of  the  line  is  in  the  NE.  or  SW.  quadrants.  In  both  cases, 

Ap  =  AlcotZ; 

Ap  =  AA  cos  L  ; 
AA  cos  L  =  AL  cot  Z, 

AA  =  AL  sec  L  cot  Z  ; 


but 

therefore, 

and 

but 

therefore, 


F  =  sec  L  cot  Z. 


(236) 


and  the  quantity  sec  L  cot  Z  is  the  longitude  factor.     It  is 
tabulated  in  Table  I  of  this  book,  the  arguments  being  L  and 


LATITUDE  FACTOR  611 

Z.  This  factor  may  be  found  in  Table  C  of  Lecky,  in  Inman's 
Tables,  and  in  Table  II  of  A.  C.  Johnson's  most  excellent  work 
on  finding  latitude  and  longitude  in  cloudy  weather.  Since 

/  —  l_j  /  may  be  found  from  the  above  tables  by  first  finding 

F 
from  them  the  value  of  F  and  taking  its  reciprocal. 

It  is  now  understood  that  if  F  is  known  and  either  AL  or 

AA  also  known,  the  other  may  be  found  from  the  expression 

AX  =  AL  X  F,  (237) 

and  also  that      AL  =  AA  X  y  ,  or  AL  =  AA  X  f.  (238) 

Rule  for  naming  AL  and  AA  when  the  azimuth  of  the  body 
is  given. — Regarding  the  azimuth  of  the  body  as  less  than 
90°  and  as  estimated  from  either  the  North  or  South  point 
of  the  horizon  towards  East  or  West,  we  have  the  following 
obvious  rule:  "If  the  change  in  latitude  AL  is  of  the  same 
name  as  the  first  letter  of  the  bearing,  the  change  in  longitude 
AA  is  of  the  contrary  name  to  that  of  the  second  letter,  and 
vice  versa"  Thus,  if  the  body  bears  S.  45  E.,  if  AL  is  8.,  AA 
is  W.;  and  if  AL  is  N.,  AA  is  E. 

INTERSECTION  BY  COMPUTATION. 

299.  We  are  now  prepared  to  find  the  intersection  of  lines 
of  position  by  computation. 

The  simplest  case  to  be  dealt  with  occurs  when  one  line 
runs  due  N".  or  S.,  as  when  a  body  is  observed  on  the  prime 
vertical,  and  another  line  runs  due  E.  or  W.,  as  when  a  body 
is  observed  on  the  meridian.  The  longitude  of  the  ship  is 
that  of  the  first  line,  the  latitude  is  the  parallel  of  the  second 
line. 


612 


NAUTICAL  ASTRONOMY 


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THE  CHORD  METHOD  613 

300.  The  next  case  is  where  the  line  of  position  (original  or 
transferred  for  run),  runs  at  an  angle  with  both  parallels 
and  meridians,  and  is  intersected  by  a  line  running  due  E. 
and  W. ;  this  last  line  being  a  parallel  of  latitude  from  the 
meridian  altitude  of  a  body. 

This  is  a  case  of  simultaneous  observations,  say  of  two 
stars;  one  on  the  meridian,  the  other  off  the  prime  vertical 
at  the  time  of  observation.  As  in  example  213  the  latitude 
from  the  meridian  observation,  being  well  determined,  is  used 
to  work  the  time  sight. 

Again,  a  case  under  this  heading  occurs  when  finding  the 
noon  position  from  an  a.  m.  observation,  a  run  to  noon,  and 
latitude  from  a  meridian  altitude  of  the  sun.  In  the  latter 
case,  there  are  two  ways  of  finding  the  intersection,  according 
as  the  a.  m.  line  is  determined  by  the  chord  or  tangent 
method. 

301.  (1)  The  chord  method. — Assume  two  latitudes,  about 
10'  each  side  of  the  D.  R.  latitude,  work  a  line  of  position,  ob- 
taining the  longitudes  corresponding  to  the  two  assumed  lati- 
tudes and  hence  two  points  of  the  line.     Divide  the  difference 
of  the  computed  longitudes  by  the  difference  of  the  two  as- 
sumed latitudes.     The  result  is  the  longitude  factor  F. 

Correct  each  position  of  the  line  for  the  run  to  noon, 
obtaining  the  corresponding  points  of  the  line  at  noon.  The 
difference  between  the  latitude  of  one  point  of  this  line,  after 
being  moved  for  the  run,  as  an  origin,  and  the  latitude  by 
meridian  altitude  is  AZ/  but  AA  =  AL  X  F. 

Knowing  which  way  the  line  of  position  runs,  the  sign  of 
application  of  AA  is  apparent.  The  result  obtained  by  apply- 
ing AA  to  the  longitude  of  the  point  taken  as  an  origin  will 
be  the  noon  longitude  by  observation. 

Rule  for  naming  AL  and  A  A  when  the  direction  of  the  line 
of  position  is  given. — Regarding  the  direction  of  the  line  as 
an  angle  less  than  90°,  and  as  estimated  from  either  the  North 
or  South  point  of  the  horizon,  towards  the  East  or  West 


614  NAUTICAL  ASTRONOMY 

point,  we  have  the  following  rule:  "If  AL  is  of  the  same 
name  as  the  first  letter  of  the  direction,  AA  is  of  the  same  name 
as  the  second  letter;  and  vice  versa."  Thus,  if  the  line  runs 
N.  30°  E.,  and  the  change  of  latitude  is  to  the  northward,  the 
change  of  longitude  will  be  to  the  eastward.  This  rule  applies 
to  the  "  chord  method,"  and  the  rule  in  Art.  298  to  the 
"  tangent  method." 

If  the  longitude  by  observation  is  desired  at  the  time  of  the 
a.  m.  sight,  having  found  the  a.  m.  line,  it  is  only  necessary 
to  run  the  noon  latitude  back  to  the  time  of  a.  m.  sight  by 
applying  the  run  in  latitude  from  sight  to  noon  backward, 
thus  getting  the  true  latitude  at  the  time  of  the  a.  m.  sight. 

The  difference  between  this  true  latitude  at  the  time  of 
a.  m.  sight  and  the  latitude  of  one  point  of  the  line  in  its 
a.  m.  position  as  an  origin,  multiplied  by  the  longitude  factor 
F,  gives  the  correction  in  longitude,  or  AA,  to  be  applied  to  the 
longitude  of  the  same  point.  The  result  will  be  the  longitude 
by  observation  at  the  time  of  the  a.  m.  sight. 

If,  to  this,  the  run  in  longitude  from  the  time  of  sight  till 
noon  is  applied,  the  result  will  be  the  longitude  at  noon  by 
observation,  which  should  agree  with  that  obtained  as  in  the 
previous  article. 

302.  (2)  The  tangent  method. — Work  up  the  dead  reckon- 
ing to  the  time  of  a.  m.  longitude  sight.  Work  the  time 
sight  with  the  D.  E.  latitude,  calling  the  resulting  longitude 
computed  longitude. 

With  the  latitude,  declination,  and  L.  A.  T.  from  the  sight, 
find  the  sun's  true  azimuth  from  the  azimuth  tables  or  an 
azimuth  diagram. 

The  azimuth  must  be  considered  as  less  than  90°  ;  so,  if  that 
from  the  tables  exceeds  90°,  estimated  from  one  pole,  use  its 
supplement  and  reckon  it  from  the  opposite  pole. 

With  the  D.  E.  latitude  and  the  sun's  azimuth,  find  from 
Table  I  (or  Table  C  in  Lecky,  Table  38,  Bowditch,  or  from 
Inman's  Tables)  the  longitude  factor  F;  write  the  value  of  F 


THE  TANGENT  METHOD  615 

in  the  form  for  work,  and  near  it  the  direction  of  the  line 
thus:  F  =  a./ ,  meaning  that  the  variation  in  longitude  for 
V  of  latitude  is  a  and  the  line  of  position  runs  NEd.  and  SWd., 
or  F  =  a.  \  in  case  the  line  runs  NWd.  and  SEd. ;  the  direction 
of  the  line  being  obtained  from  the  bearing  (regarded  as  less 
than  90°)  ~by  changing  either  letter  of  the  bearing  and  taking 
the  complement  of  the  angle. 

To  this  D.  E.  latitude  and  computed  longitude  apply  the  run 
to  noon,  obtaining  at  noon  a  D.  E.  latitude  and  a  computed 
longitude.  The  line  at  noon  remains  parallel  to  its  direction 
at  time  of  sight,  and  F  has,  of  course,  the  same  value. 

With  the  computed  longitude,  work  the  meridian  altitude 
sight  and  find  latitude  at  noon  by  observation. 

The  difference  between  the  latitude  at  noon  by  D.  E.  and 
by  observation  is  AL,  or  the  error  in  latitude.  As  before, 
AA  =  AL  X  F. 

In  the  absence  of  Table  I,  the  correction  AA  may  be  found 
thus:  Enter  Table  2  of  Bowditch's  Useful  Tables  with  the 
complement  of  the  bearing  as  a  course,  find  AL  in  the  latitude 
column,  and  take  the  corresponding  departure  from  the  de- 
parture columns.  This  departure,  converted  into  difference 
of  longitude,  will  be  AA. 

Having  found  AA,  its  sign  of  application  may  be  found 
from  rule  of  Art.  298,  if  the  azimuth  of  the  sun  is  considered ; 
or,  rule  of  Art.  301,  if  the  direction  of  the  line  is  considered. 
Apply  AA  to  the  computed  longitude  for  noon,  the  result  will 
be  the  longitude  at  noon  by  observation. 

Should  it  be  desired  to  find  the  true  longitude  at  time  of 
sight,  run  the  noon  latitude  back  to  the  time  of  the  a.  m. 
sight  by  applying  the  run  in  latitude  from  sight  to  noon  back- 
wards, getting  the  true  latitude  at  time  of  sight.  The  differ- 
ence between  this  latitude  and  the  D.  E.  latitude  at  sight  is 
AL;  then  AA  =  AL  X  F  is  the  correction  in  longitude  to  be 
applied  to  the  computed  longitude  at  time  of  sight  to  give  the 
longitude  by  observation  at  that  time. 


616 


NAUTICAL  ASTRONOMY 


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MUTUAL  CORRECTION  METHOD  619 

When  one  Observation  is  of  a  Body  within  45°  of  the  Prime 

Vertical  and  the  other  of  a  Body  within  15°  of 

the  Meridian. 

303.  The  mutual  correction  method. — This  method  applies 
to  the  following:  (1)  A  case  of  simultaneous  observations 
in  which  one  body  is  observed  near  the  prime  vertical  for  time 
and  one  near  the  meridian  for  latitude;  (2)  a  case  of  double 
altitudes  of  the  sun  with  an  intervening  run — the  first  alti- 
tude observed  within  45°  of  the  prime  vertical  for  time,  the 
second  altitude  observed  within  15°  of  the  meridian  and 
worked  as  a  "  reduction  to  the  meridian  "  sight. 

Having  determined  the  first  line,  by  either  the  chord  or 
tangent  method,,  and  corrected  the  coordinates  of  one  point 
for  the  run,  and  having  found  the  value  of  F± ,  it  is  not  un- 
usual to  consider  the  latitude  obtained  by  "reduction  to  the 
meridian"  (using  the  computed  longitude  at  the  instant  of 
observation  in  finding  H.  A.),  as  sufficiently  exact  for  all 
practical  purposes,  and  for  this  latitude  to  find,  as  in  examples 
214  and  215,  the  longitude  of  fix.  Then  the  noon  position  is 
found  by  applying  to  the  latitude  and  longitude  of  fix  the 
run  from  the  time  of  the  second  observation  till  noon. 

However,  as  the  body  at  the  second  observation  is  not  on 
the  meridian  and  the  resulting  line  of  position  is  not  East 
and  West  in  direction,  and  as  the  sight  is  worked  with  a 
longitude  which  may  be  in  error,  the  latitude  obtained  may  be 
in  error. 

For  more  precise  results,  the  "  mutual  correction  method  " 
may  be  used,  correcting  the  longitude  from  the  time  sight  by 
the  formula  AX±  =  Al^  X  ^ , 

in  which  ALj.  is  the  difference  between  the  latitude  of  a  posi- 
tion point  in  the  first  line  corrected  for  the 
run  to  second  observation  and  the  computed 
latitude  at  the  second  observation; 


620  NAUTICAL  ASTRONOMY 

FI  is  the  longitude  factor  of  the  first  line ; 
and  AA±  is  the  correction  to  be  applied  to  the  computed 
longitude  at  the  second  observation  to  give  the 
longitude  of  fix. 

Then  the  latitude  from  the  "reduction  to  the  meridian" 
sight  should  be  corrected  by  the  formula  AL2  =  AA±  -=-  F2 ; 
in  which  F2  is  the  longitude  factor  of  the  second  line  ob- 
tained from  Table  I,  knowing  the  latitude  and 
sun's  azimuth  at  second  observation ; 
and  AL2  is  the  correction  to  be  applied  to  the  computed 
latitude  at  the  second  observation  to  give  the 
latitude  of  fix. 

This  method  of  "mutual  correction "  is  applicable  only 
where  one  observation  is  a  time  sight  and  one  a  sight  near 
the  meridian,  the  latitude  from  this  latter  sight  being  nearly 
correct. 

The  following  rules  are  given  for  tto  second  case  under 
this  heading,  that  of  double  altitudes  of  the  sun;  modifica- 
tions necessary  to  make  them  fit  the  first  case,  that  of  simul- 
taneous observations,  will  be  apparent. 

Rules. — Work  the  time  sight  by  either  the  chord  or  tan- 
gent method;  using  the  D.  K.  latitude  in  the  latter  case,  or 
assuming  latitudes  about  10'  each  side  of  the  D.  K.  latitude 
in  the  former  case. 

Find  the  longitude  factor  of  first  line  F^ . 
To  the  coordinates  of  one  position  point  of  the  first  line 
apply  the  run  to  the  instant  of  second  observation  and  obtain 
a  D.  E.  latitude  and  a  computed  longitude;  and,  with  this 
longitude,  work  the  second  sight  by  the  "reduction  to  the 
meridian"  method,  obtaining  a  computed  latitude. 

With  the  computed  latitude  and  azimuth  at  the  second  ob- 
servation, find  the  longitude  factor  of  the  second  line  F2 . 

Take  AL±  equal  to  the  difference  between  the  D.  E.  latitude 
and  the  computed  latitude  at  the  second  observation  and  find 


MUTUAL  CORRECTION  METHOD  621 


AAj  =  AL±  X  FI',  apply  AAj.  to  the  computed  longitude  at 
the  second  observation  and  obtain  the  longitude  of  fix. 

From  AA±  find  AL2  =  AAi  -f-  F2  ;  apply  AL2  to  the  com- 
puted latitude  at  the  second  observation  and  obtain  the  lati- 
tude of  fix. 

Ex.  216.  —  About  7.45  a.  m.,  January  1,  1905,  from  an  ob- 
servation of  the  sun  in  latitude  16°  21'  34"  N.  by  D.  E.  found 
the  computed  longitude  to  be  63°  15'  30"  W.  True  azimuth 
of  sun  =  ZN  —  I20f  °.  Ran  thence  till  llh  40m  a.  m.  315° 
(true)  30.7  miles  when,  by  the  "reduction  to  the  meridian" 
method,  the  latitude  was  found  to  be  16°  45'  04"  K  True 
azimuth  of  sun  ZN  =  172°.8.  Ean  thence  till  noon  315° 
(true)  2.7  miles.  Required  the  noon  position. 

D.  R.  between  sights. 


True  Course.     I     Distance.          Diff.  Lat. 


Dep.        I     Diff.  Long. 
21.7  W  22'.64  W 


315°  I  30.7  I         21.7  N 

o    /    ii  o    i    it 

Lat.  by  P.  R.  at    7.45  a.m.  16  21  34  N       Long,  computed  at  7.45a.m.  63  15  30    W 
£>iff.  of  Lat.  to  11.40a.m.       21  42  N       Diff.  Long,  to  11.40a.m.       22  38.4  W 

Lat.  by  D.  R.  at  11.40  a.m.  16  43  16  N       Computed  Long,  at  11.40  a.m.  63  38  08.4  W 
Lo  =  16  32  25N 

o    /   n 

At  11.40  a.  m.  computed  Lat.  16  45  04  N 
At  11.40  a.  m.  Lat.  by  D.  R.  16  43  16  N 
A  LI  =  I'.S  N  =  1  48  N 

To  correct  the  longitude  from  the  a.  m.  time  sight. 

Lat.  16°.36  N,  *1=S  59J°  E,  J\  =  .62      Computed  Long,  at  11.40  a.  m.  63  38  08.4  W 
AX1=AL1Xl;Ti=1.8X.62=lM16      AAj  j  07     E 

AA1=1M16=1'  07"  Long,  of  fix  11.40  a.  m.  63  37  01.4  W 

Aij  is  northerly,  Zl  is  southward  and  eastward;  therefore,  A^!  is  easterly. 


To  correct  the  latitude  obtained  from  the  sight  near  noon. 

Lat.l6°.75N,^2=S7°.2E,  F2=8.29       Computed  Lat.  at  11.40  a.  m.       164504    N 


AL2=OM3=7".8  Lat>  of  flx  1L40  a>  m>  16  45  11.8  N 

AAj  is  easterly,  Z^  southward  and  eastward;  therefore,  Ai2  is  northerly. 

To  find  the  noon  position. 

D.  R.  11.40  a.  m.  to  noon. 


True  Course. 
315° 

Lat.  of  flx  11.40  a.  m.    16 
Diff.  of  Lat.  to  noon 
Lat.  in  at  noon             16 

1     Distance. 

1            2.7 
i     it 
45  11.8  N 
1  54     N 
47  06     N 

1     Diff.  Lat.     1        Dep. 
I         1.9  N         1        1.9  W 

Long,  of  fix  11.40  a.  m. 
Diff.  of  Long,  to  noon 
Long,  at  noon 

o     /    n 
63  37  01.4  W 
200     W 
633901     W 

622 


JNAUTICAL  ASTRONOMY 


304.  To  determine  the  intersection  of  two  lines  running  at 
an  angle  with  both  meridians  and  parallels,  when  position 
points  having  a  common  latitude  are  known,  one  for  each 
line. 

Two  lines  with  position  points  on  a  common  parallel  may 
be  considered  when  we  have  simultaneous  observations  of 
two  bodies  favorably  situated  for  finding  time,  the  ZN  of 
each  being  from  45°  to  135°  or  225°  to  315°;  also  when  a 
line  from  a  time  sight  is  combined  with  one  from  a  </>"</>' 


Dissimilar" 


M  C 

FIG.  133. 


sight  previously  taken,  the  computed  latitude  from  the  latter, 
after  correction  for  the  run  in  the  interval,  being  used  in  the 
time  sight. 

Two  cases  occur  under  this  head:  (1)  when  both  lines  are 
in  the  same  or  opposite  quadrants,  being  then  called  similar; 
(2)  when  the  two  lines  run  in  adjacent  quadrants,  being  then 
called  dissimilar. 

The  chord  method. — Let  A  and  B  be  two  points  of  a  line 
determined  by  assuming  latitudes  L±  and  L2 ,  and  C  and  D 
two  points  of  a  second  line  having  the  same  coordinates;  the 
two  lines  being  the  results  of  simultaneous  observations  of 
two  different  bodies;  or,  of  observations  taken  at  different 
times,  whether  of  the  same  or  different  bodies,  one  line,  how- 
ever, being  brought  up  to  the  time  of  the  second  observation 
for  the  run  in  the  interval.  Let  \\  and  A'2  be  the  longitudes 
of  A  and  B;  \'\  and  A"2 ,  of  C  and  D  respectively. 


ASSUMING  LATITUDES  623 

Let  AB  be  the  first  line,  F±  its  longitude  factor  found  as  in 
Art.  298.  Then,  if  P  is  the  point  of  intersection,  AL  =  PM, 
and  the  corresponding  difference  of  longitude  from  A  is 
AA±  =  AM;  therefore,  AAt  =  AL  X  FI . 

If  CD  is  the  second  line,  its  longitude  factor  is  F2  and,  in 
the  same  way  as  above,  CM  —  AA2  and  AA2  =  AL  X  F2 . 

In  both  figures,  AC  is  the  known  difference  of  longitude  of 
both  lines  for  one  assumed  latitude.     In  Fig.  132, 
AC  =  AM  —  CM  =  PM  (F±  —  F2), 
or  AM  —  CM  =  AL  (F±  —  F2) ; 
but  AA±  —  AM  =  AL  X  F± , 
and  AA2  =  CM  =  AL  X  F2 . 

In  this  figure,  where  the  lines  are  similar,  both  AA±  and  AA2 
are  applied  in  the  same  direction,  both  East  or  both  West,  so 
as  to  make  the  resulting  longitudes  the  same. 

Then,  knowing  the  name  of  the  correction  in  longitude  for 
either  line  and  the  direction  of  the  line,  the  name  of  the  cor- 
rection in  latitude  is  apparent,  or  is  found  by  the  rule  of 
Art.  301,  in  which  the  direction  of  the  line,  instead  of  the 
bearing  of  the  body,  is  considered. 

In  Fig.  133, 

AC  =  AM  +  CM  =  PM  (F±  +  F2)  =  AL  (F±  +  F2) 
and  AA±  =  AM  =  AL  X  F±  and  AA2  =  CM  =  AL  X  F2  • 

In  this  figure,  where  the  lines  are  "  dissimilar"  AAt  and 
AA2  are  applied  in  the  opposite  directions,  westerly  to  the 
more  easterly  longitude  and  easterly  to  the  more  westerly  lon- 
gitude, so  as  to  make  both  resulting  longitudes  the  same,  which 
must  be  the  case;  otherwise  an  error  has  been  made.  Then, 
knowing  the  name  of  the  correction  in  longitude  for  either 
line,  and  the  direction  of  the  line,  the  name  of  the  correction 
for  latitude  is  found.  Apply  the  correction  in  latitude  to  the 
latitude  of  the  parallel  used  as  origin  to  find  the  latitude  of 
fix. 


624 


NAUTICAL  ASTRONOMY 


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626  NAUTICAL  ASTRONOMY 

Rules  for  the  chord  method  using  a  common  latitude. — 

(1)  Find  the  longitude  factors  F±  for  the  first  line,  and 
F2  for  the  second  line  (Art.  298). 

(2)  Divide  the  difference  of  the  longitudes  computed  for 
each  line  with  a  given  common  latitude  by  the  difference  or 
sum  of  the  longitude  factors,  according  as  the  lines  are  "  simi- 
lar "  or  "  dissimilar!'     The  result  is  the  correction  in  lati- 
tude, A.L. 

(3)  The  correction  in  latitude   (AL),  multiplied  by  the 
longitude  factor  of  each  line  (F±  and  F2),  gives  the  correc- 
tion in  longitude  for  that  line  (AAX  for  first  line,  AA2  for  sec- 
ond line).     For  "similar"   lines,  apply  the  corrections  in 
longitude  the  same  way  to*  the  computed  longitudes  so  as  to 
make  the  resulting  longitudes  the  same  and  obtain  the  longi- 
tude of  the  fix.     If  they  do  not  come  out  the  same,  a  mistake 
has  been  made.     This  fact  serves  as  a  check  on  the  work. 

(4)  Knowing  the  name  of  the  longitude  correction  and  the 
direction  of  the  line,  find  by  the  rule  (Art.  301),  the  name  of 
the  latitude  correction.     Apply  the  correction  in  latitude  to 
the  given  common  latitude  and  find  the  latitude  of  fix. 

In  the  above  example,  the  common  latitude  was  taken  as 
20°  30'  S.  The  lines  are  "dissimilar"  and  the  correction 
AX1?  the  longitude  correction  to  the  coordinate  of  the  first  line, 
is  marked  West  because  the  position  point  of  that  line  has 
the  most  eastern  computed  longitude ;  AX2 ,  the  longitude  cor- 
rection to  the  coordinate  of  the  second  line,  is  marked  East 
because  the  position  point  of  that  line  has  the  most  western 
longitude.  AL  is  marked  S.  as  per  rule  in  last  paragraph  of 
Art.  298. 

305.  The  tangent  method.— -Fig.  132  and  Fig.  133  apply 
in  this  case.  Only  one  latitude,  and  that  by  T>.  R.,  is  used. 
The  longitude  factor  of  each  line  is  gotten  from  Table  I,  as 
explained  in  Art.  298,  and  the  direction  of  each  line  is  ob- 
tained as  shown  in  Art.  297.  From  this  point  on,  the  mode 
of  procedure  and  the  rules  of  the  chord  method  apply. 


THE  TANGENT  METHOD 


627 


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NAUTICAL  ASTRONOMY 


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ASSUMING  LONGITUDES 


629 


306.  To  determine  the  intersection  of  two  lines  running  at 
an  angle  with  both  meridians  and  parallels,  when  position 
points  having  a  common  longitude  are  known,  one  for  each 
line. 

In  previous  articles  we  have  considered  a  position  point  in 
each  line  with  a  common  latitude.  It  may  be  necessary  to  con- 
sider two  lines  with  position  points  on  a  common  meridian,  as 
in  the  case  of  two  simultaneous  observations  worked  for  lati- 
tude by  the  <£"<£'  method;  or,  in  the  case  of  a  line  from  a 
</>"<£'  sight  combined  with  one  from  a  time  sight,  the  com- 


FIG.  134. 


FIG.  135. 


puted  longitude,  from  the  latter,  after  correction  for  the 
run  in  the  interval,  being  used  in  the  </>"<£'  sight. 

The  latitude  factor. — In  this  case,  instead  of  using  the 
variation  in  longitude  for  1'  of  latitude,  or  the  factor  ]?,  we 
use  the  variation  in  latitude  for  1'  of  longitude,  or  the  latitude 

factor  /  (which  equals  —  ). 

The  chord  method.— Let  AB  and  CD  (Fig.  134)  and  (Fig. 
135)  be  two  lines  of  position  obtained  by  working  sights  with 
the  same  assumed  longitudes,  the  longitudes  of  the  meridians 
of  AC  and  BD;  or,  let  AB  be  a  first  line  moved  for  the  run 
between  observations;  A  and  B  being  the  coordinates  of  the 


630  NAUTICAL  ASTRONOMY 

first  line  at  the  instant  of  the  second  observation,  the  longi- 
tudes of  A  and  B  are  used  to  work  the  sight  for  the  second 
line  CD.  Let  L\  and  L'2  be  the  latitudes  of  A  and  B;  L\ 
and  L"2 ,  of  C  and  D  respectively. 

AB  being  the  first  line,  /±  (found  as  in  Art.  298)  is  its 
latitude  factor;  then,  if  P  is  the  point  of  intersection  of  AB 
with  the  second  line,  taking  the  meridian  of  AC  as  origin, 
we  have  AA  =  PM  and  the  corresponding  AL  from  parallel 
of  A  is  AM,  but  AM  =  PM  X  fa  therefore,  AL±  =  AA  X  /± . 

If  CD  is  the  second  line,  its  latitude  factor  is  fz ,  and,  in  the 
same  way  as  above,  CM  =  PM  X  /2  >  or  AL2  —  AA  X  fz  • 

In  both  figures,  AC  is  the  known  difference  of  latitude  of 
the  position  points  of  the  two  lines  on  the  common  meridian 
AC. 

In  Fig.  134,  AC  —  AM  —  CM  =  PM(f±  --  f2),  or  the 
difference  of  latitude  AC  =  AA  (f±  —  f2),  then  AM  =  AL± 
—  A  A  X  f  i  and  CM  =  AL2  =  AA  X  f  2  • 

In  this  figure,  where  the  lines  are  "similar'3  both  correc- 
tions to  the  latitude,  AL±  and  AL2 ,  are  applied  in  the  same 
way,  either  to  N.  or  to  S.  so  as  to  make  the  resulting  latitudes 
the  same. 

Knowing  the  name  of  the  correction  in  latitude  for  either 
line,  the  name  of  the  correction  in  longitude  for  the  same  line 
is  apparent,  or  is  obtained  by  the  rule  of  Art.  301. 

In  Fig.  135,  AC  =  AM  +  CM  =  PM(f±  '-f  f£),  or  the 
known  difference  of  latitude  AC  =  AA(f±  +  f2) ;  then  AM 
=  AL±  =  AA  X  f i  and  CM  =  AL2  =  AA  X  f 2  • 

In  this  figure,  where  the  lines  are  "  dissimilar/'  the  cor- 
rections for  latitude  are  applied  in  the  opposite  directions, 
northerly  to  the  more  southern  latitude  and  southerly  to  the 
more  northern  latitude,  so  as  to  make  both  resulting  latitudes 
the  same.  The  name  of  the  correction  in  longitude  is  found 
by  rule  (Art.  301).  The  correction  in  longitude,  applied 


ASSUMING  LONGITUDES  631 

with  its  proper  sign  to  the  longitude  of  the  common  meridian, 
will  give  the  longitude  of  fix. 
Rules  for  chord  method,  using  a  common  longitude. — 

(1)  From  the  coordinates  of  the  two  points  of  each  line, 
find  the  latitude  factor  f^  for  the  first  line,  and  f2  for  the  sec- 
ond line. 

(2)  Divide  the  difference  of  the  computed  latitudes  of  the 
position  points  on  the  common  meridian  'by  the  difference  or 
sum  of  the  latitude  factors,,  according  as  the  lines  are  "simi- 
lar "  or  ee  dissimilar"     The  result  is  the  correction  in  longi- 
tude. 

(3)  The  correction  in  longitude,  multiplied  by  the  latitude 
factor  of  a  line,  gives  the  correction  in  latitude  to  be  applied 
to  the  latitude  of  that  line's  position  point  on  the  common  me- 
ridian.    For  "  similar  "  lines,  the  latitude  corrections  are  ap- 
plied the  same  way,  for  "  dissimilar  "  lines  the  opposite  way, 
to  the  computed  latitudes,  so  as  to  malce  the  resulting  latitude 
the  same  and  to  give  the  latitude  of  fix. 

(4)  Knowing  the  name  of  the  latitude  correction,  and  the 
direction  of  the  line  of  position  corresponding,  find  by  the 
rule  (Art.  301)  the  name  of  the  longitude  correction.    Apply 
this  with  its  proper  sign  to  the  longitude  of  the  common  me- 
ridian; the  result  will  be  the  longitude  of  fix. 


632 


NAUTICAL  ASTRONOMY 


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THE  TANGENT  METHOD  635 

307.  The  tangent  method.  —  The  above  example  (219)  may 
be  worked  by  the  tangent  method  thus:  Let  the  first  sight 
be  worked  with  the  D.  E.  latitude  40°  24'  N.,  giving  a  com- 
puted longitude;  find  from  the  azimuth  tables  the  sun's  true 
bearing  Z±  at  the  first  observation,  and  from  Table  I  the 
value  /±  . 

Let  this  D.E.  latitude  and  the  resulting  computed  longitude 
be  corrected  for  the  run  of  the  ship  between  observations,  thus 
obtaining  a  position  point  Lr,  A'. 

Let  the  $"$'  sight  be  worked  with  this  corrected  longitude 
A',  the  resulting  computed  latitude  being  L".  Find  from  the 
azimuth  tables  the  sun's  true  bearing  Z2  at  the  second  obser- 
vation and  from  Table  I  the  value  /2;  the  azimuths  in  both 
cases  being  taken  less  than  90°. 

In  example  219,  the  lines  run  in  the  same  quadrant  and  are 
"  similar  "  ;  therefore, 

Diff  .  of  latitudes  L"  ~  L' 


_ 


Diff.  of  Lat.  factors        / 


AL±  =  AA  X  A  and  L  =  Lat.  of  fix  =  L'  +  AL±  ,  "I          , 
AL2  =  AA  X  f2  and  L  =  Lat.  of  fix  =  L"  —  AL2  .  J  ° 
Then  mark  AA  with  its  proper  sign  and  apply  it  to  X'  ;  the  re- 
sult A  is  the  longitude  of  fix. 


CHAPTEE  XX. 
THE  NEW  NAVIGATION. 

308.  The  method  of  treating  Sumner  lines  to  be  described 
in  the  following  pages  is  known  as  the  method  of  Marcq  Saint- 
Hilaire,  and  is  considered  in  various  text-books  under  the 
head  of  "  The  New  Navigation."  However,  there  is  nothing 
new  about  it — the  French  have  used  the  method  for  years. 

Eeferring  to  Arts.  281-284  and  Fig.  121,  it  is  seen  that 
for  a  given  instant  an  observed  heavenly  body  is  in  the  zenith 
of  some  place  on  the  earth's  surface,  which  is  the  pole  of  a 
system  of  circles  of  equal  altitude  called  also  parallels  of 
altitude.*  For  a  given  dead  reckoning  position,  this  body  has 
a  parallel  of  altitude,  and  the  altitude  corresponding  may  be 
found  by  computation. 

If  from  the  ship's  position  at  the  given  instant,  the  meas- 
ured altitude  of  the  same  heavenly  body  differs  from  the  com- 
puted altitude,  the  true  position  of  the  ship  is  not  on  the 
parallel  of  altitude  passing  through  the  dead  reckoning  posi- 
tion but  on  another  one  of  the  same  system ;  distant  from  it 
on  the  great  circle  passing  through  the  dead  reckoning  posi- 
tion of  the  ship  and  the  body,  the  number  of  sea  miles  equal 
to  the  number  of  minutes  of  arc  in  the  difference  between  the 
measured  and  computed  altitudes,  and  in  a  direction  towards 
or  from  the  observed  heavenly  body,  according  as  the  measured 
altitude  is  greater  or  less  than  that  by  computation  ;  therefore, 
to  determine  one  point  of  the  circle  of  equal  altitudes,  which 
circle  is  the  locus  of  the  ship's  possible  positions  at  the  given 


THE  NEW  NAVIGATION 


637 


instant,  it  is  only  necessary  to  lay  off,  in  the  proper  direction 
from  the  ship's  position  by  D.  E.,  a  great  circle  distance 
equal  to  the  above-mentioned  difference  of  altitude  called  the 
"  altitude  difference/'  The  point  thus  determined  is  one 
"  position  point "  and  the  circle  of  altitude  through  it  is  the 
required  "  line  of  position/' 

Let  Fig.  136  represent  a  projection  of  a  heavenly  body  8 
on  the  plane  of  the  horizon  of  its  geographical  position 
(Art.  281),  showing  A,  the 
D.  E.  position  of  a  ship 
from  which  an  altitude  of 
the  body  S  has  been  ob- 
served, then  AS  is  the  great 
circle  direction  of  the  body 
8-  from  A  and  HA  is  the 
computed  true  altitude  of  8 
for  that  place  and  instant  of 
observation.  If  the  meas- 
ured true  altitude  is  also 
equal  to  HA,  the  ship's  real 
position  is  on  a  circle  of 
altitude  whose  radius  is  SA ;  however,  if  the  measured  true 
altitude  is  not  HA  but  HB  (represented  in  the  figure  as 
>  HA),  then  AB  is  the  "altitude  difference,"  IBcd  is  the 
circle  of  altitude  passing  through  the  place  of  observation, 
and  B  is  one  point  of  this  circle  which  has  certain  attributes 
that  make  it  very  prominent  in  the  methods  of  the  so-called 
"  New  Navigation." 

This  point  is  always  nearer  to  the  real  position  of  the  ship 
than  the  D.  E.  position,  except  when  coincident  with  it,  as 
may  be  seen  by  reference  to  Fig.  136,  in  which  ahc,  a  circle 
described  about  A  as  a  center  with  a  radius  equal  to  the  pos- 
sible error  and  called  "  a  circle  of  error,"  includes  the  ship's 
position  which,  being  also  on  the  circle  of  altitude  bBcdf 


FIG.  136. 


638  NAUTICAL  ASTRONOMY 

must  be-  on  the  arc  IBc.  Since  ABS  is  perpendicular  to  this 
arc  at  its  middle  point  B  it  is  evident  that  B  is  nearer  than 
A  to  the  ship's  real  position,  and  as  B  occupies  the  mean  of 
the  probable  positions,  it  is  less  likely  to  be  in  error  than 
other  points  of  the  arc. 

The  circle  of  altitude  might  be  drawn  on  a  globe  if  the  lat- 
ter should  have  sufficient  dimensions  (Art.  283)  ;  or  it  might 
be  practicable  to  draw  it  on  a  Mercator  chart  in  the  case  of  a 
body  of  small  declination  observed  at  a  high  altitude,  in  other 
words,  in  case  of  many  observations  made  in  the  tropics,  by 
first  locating  the  body's  geographical  position  and  then  draw- 
ing a  circle  from  that  point  as  a  center  with  a  radius  equal  to 
the  body's  observed  true  zenith  distance  (Arts.  281-286). 
The  point  B  would  then  be  determined  by  construction  at  the 
intersection  of  this  circle  and  the  great  circle  passing  through 
the  D.  E.  position  of  the  observer  and  the  body's  geographical 
position. 

The  computed  point. — As  the  use  of  a  globe  is  impracti- 
cable, and  since  circles  of  altitude  may  be  represented  by 
circles  on  a  Mercator  chart  only  under  special  circumstances, 
it  is  ordinarily  necessary  to  find  by  computation  the  co-ordi- 
nates of  the  point  B  through  which  the  circle  of  altitude 
passes,  therefore,  in  the  practice  of  the  "  New  Navigation," 
this  point  becomes  the  first  desideratum  and  will  be  referred 
to  hereafter  as  the  "  computed  point."  Having  determined 
the  "  altitude  difference,"  this  point  may  be  found  with  suffi- 
cient accuracy  in  practical  navigation  by  laying  off  this  dif- 
ference on  a  loxodrome  through  the  D.  E.  position  instead  of 
along  the  great  circle  bearing,  the  error  produced  by  this  sub- 
stitution, owing  to  the  small  size  of  the  "  altitude  difference," 
being  inappreciable,  even  under  the  most  unfavorable 
conditions. 

Line  of  position. — Since  the  D.  E.  position  so  limits  that 
portion  of  the  circle  of  altitude  on  which  the  observer  may 


THE  NEW  NAVIGATION  639 

be,  it  is  necessary  in  practice  to  consider  only  a  small  arc 
and  this  will  not  differ  materially  within  certain  limits  from 
a  straight  line  drawn  through  the  computed  point  at  right 
angles  to  the  body's  bearing  regarded  as  a  loxodrome,  except 
when  the  body  observed  is  very  near  the  zenith,  the  limits  of 
coincidence  depending  on  the  value  of  the  altitude  difference 
as  well  as  the  altitude  of  the  observed  body. 

Formulae. — The  computed  altitude  and  azimuth  may  be 
found  from  the  formulae  (159),  the  form  for  arrangement  of 
work  in  finding  h  and  Z  being  as  shown  in  the  solution  of 
examples  157  and  158. 

Or,  as  is  the  better  method,  the  computed  altitude  may  be 
found  from  the  formula 

sin  h  =  cos  (L  ~  d)  —  2  cos  L  cos  d  sin2  %t       (239) 

and  the  true  azimuth  Z  may  then  be  taken  from  the  azimuth 
tables  (Art.  221),  from  an  azimuth  diagram,  or  found  from 
a  simultaneous  compass  bearing  corrected  for  variation  and 
deviation,  provided  the  conditions  are  such  as  to  admit  of  an 
accurate  bearing  being  taken. 

For  a  body  observed  on  the  meridian  formula  (239)  re- 
duces to  sin  h  =  cos  (L~  d)  or  z  =  L  ~  d,  the  same  formula 
as  in  the  direct  method  (240),  and  in  this  case  the  "New 
Navigation  "  has  no  advantages. 

Should  the  navigator  be  provided  with  Inman's  Tables,  or 
tables  of  haversines  and  log  haversines,  the  following  formulae 
,  obtained  from  (209)  and  (210)  by  substituting  (L~d)  for  z0, 
•are  strongly  recommended: 

haver  z  =  haver  (L~d)-\-  haver  0,1 
where  z  =  90°  —  li  and  0  is  defined  by  (240) 

haver  6  =  cos  L  cos  •?  haver  t. 

A  line  determined  by  the  Marcq  Saint-Hilaire  method  may 
be  used  in  the  same  way  as  one  determined  by  any  other 


640  NAUTICAL  ASTRONOMY 

method  (Art.  291) ;  it  may  be  combined  to  determine  a  fix 
with  a  terrestrial  bearing  or  with  a  line  found  from  a  sight 
worked  by  some  one  of  the  direct  methods. 

Conditions  of  observation. — Owing  to  errors  of  refraction 
at  low  altitudes,  and  to  the  small  limits  within  which  the 
circle  of  altitude  and  its  tangent  at  the  computed  point  are 
coincident  at  very  high  altitudes  as  well  as  the  practical  diffi- 
culties of  observation  under  such  circumstances,  it  is  desirable 
that  heavenly  bodies  be  observed,  if  possible  at  altitudes  not 
less  than  10°  nor  greater  than  86°. 

Advantages  of  the  Marcq  Saint-Hilaire  method. — The  great 
advantage  of  this  method  of  obtaining  a  line  lies  in  the  fact 
that  since  the  formulce  make  it  available  practically  without 
limitations  as  to  azimuth,  altitude,  or  hour  angle,  it  furnishes 
one  method  equally  applicable  to  all  conditions,  whether  these 
conditions  would  otherwise  require  the  formulce  of  a  time- 
sight,  a  <J>"<f>'  sight,  or  that  of  a  body  observed  near  the  merid- 
ian. Except  when  finding  latitude  by  meridian  altitude,  by 
reduction  to  the  meridian,  or  by  Polaris,  or  when  finding 
longitude  by  the  time-sight  (tangent)  method,  the  process  of 
solution  above  described  is  simpler  than  any  other  method 
for  either  latitude  or  longitude. 

Rule  for  the  determination  of  a  single  line. — With  the 
latitude  and  longitude  of  the  given  D.  E.  position,  compute 
the  true  altitude  of  a  heavenly  body  for  the  instant  at  which 
the  observed  true  altitude  is  known,  or  will  be  known,  and  find 
from  the  azimuth  tables  the  body's  true  azimuth  Z  for  the 
same  instant.  Subtract  the  computed  true  altitude  from  the 
observed  true  altitude,  calling  the  remainder  the  "  altitude 
difference  "  and  designating  it  by  the  letter  a.  Then  run  by 
dead  reckoning,  or  lay  down  on  the  chart  from  the  assumed 
D.  E.  position,  the  distance  of  a  sea  miles  on  a  Mercator  course 
equal  to  the  observed  body's  azimuth  if  the  observed  altitude 
is  the  greater;  on  a  course  equal  to  the  azimuth  -j-1800,  if 


THE  NEW  NAVIGATION  641 

the  observed  altitude  is  less  than  that  by  computation.  The 
point  thus  determined  is  the  computed  point,  and  a  line 
(which  may  be  drawn  on  the  chart)  through  this  point  at 
right  angles  to  the  body's  bearing  line  will  be  the  line  of  posi- 
tion. In  Fig.  137,  AB  is  the  "altitude  difference,"  B  the 
computed  point,  and  BB'  the  line  of  position  from  an  obser- 
vation of  the  body  St  made  at  the  D.  R.  position  A. 

Double  altitudes. — If  having  determined  the  computed 
point  B  of  one  line  BB'  (Fig.  137),  the  run  of  the  ship  is 
laid  off  from  B  to  C,  at  which  point  another  altitude  of  the 
same,  or  of  a  different  body,  bearing  in  the  direction  C82  is 
both  observed  and  computed,  the  hour  angle  from  the 
meridian  of  C  being  used  in  the  computation,  then  a  second 
line  of  position  DD'  may  be  obtained  by  laying  off  the  "  alti- 
tude difference  "  at  the  second  observation  as  before  explained, 
the  point  C  being  the  D.  R.  position  and  D  the  computed 
point  of  this  second  line.  The  intersection  of  DD'  with  CO' 
drawn  through  C  parallel  to  the  first  line  (CCf  being  the 
first  line  transferred  for  the  run  of  the  ship  during  the  inter- 
val between  observations)  will  be  the  fix  F.  These  lines  and 
the  intervening  run  may  be  laid  down  on  a  Mercator  chart 
(Fig.  137)  and  the  fix  found  by  construction  (see  Art.  292). 

Anticipating  the  work. — Should  the  navigator,  assuming 
the  position  A  for  a  later  instant,  compute  the  true  h  and  Z 
of  a  heavenly  body  to  be  observed  at  that  future  time  and 
place,  he  may  immediately  plot  that  position  on  the  chart 
and  lay  down  thfe  body's  bearing  line;  then  there  will  be 
nothing  to  do  but  to  lay  off  the  "altitude  difference"  and 
draw  in  the  line  of  position  when  the  actual  altitude  has 
been  observed  at  the  predetermined  instant  of  G.  M.  T.  The 
fact  that  the  preliminary  computation  may  be  made,  perhaps 
hours  in  advance  of  the  actual  measurement  of  the  altitude, 
makes  the  Saint-Hilaire  method  most  useful  to  any  navigator 
desiring  to  anticipate  his  work.  If  for  any  reason  the  obser- 


642 


XAUTICAL  ASTRONOMY 


FIG.  137. 


THE  XEW  NAVIGATION  643 

vation  should  not  be  made  at  the  exact  G.  M.  T.  used  in  the 
computation,  the  line  plotted  as  above  should  be  shifted  so 
as  to  allow  for  the  error  (Art.  293),  remembering  that  if  the 
G.  M.  T.  of  observation  is  greater  than  that  of  computation 
the  line  must  be  shifted  to  the  westward,  otherwise  to  the 
eastward.  The  same  result  may  be  accomplished  by  laying 
off  the  "  altitude  difference  "  from  a  new  position  point  found 
by  so  altering  the  longitude  of  the  assumed  position  (to  the 
westward  or  eastward  as  indicated  above)  that  the  hour  angle 
corresponding  to  the  G.  M.  T.  of  observation  may  be  the 
same  as  that  used  in  the  solution,  thus  making  the  computa- 
tion for  the  altitude  still  hold  good  (see  Ex.  222  and  Fig. 
145)..  For  any  probable  difference  between  the  G.  M.  T.  of 
observation  and  that  of  computation  the  changes  in  the  ele- 
ments used  for  the  observed  body  would  be  so  slight  as  to 
produce  only  an  inappreciable  error  in  the  results.  So  long 
as  the  longitude  is  changed  as  above  to  make  the  hour  angle 
the  same  for  the  G.  M.  T.,  both  of  observation  and  computa- 
tion, no  other  adjustment  will  be  necessary,  as  for  instance 
for  any  change  in  the  assumed  position  due  to  fleet  maneu- 
vers. A  line  from  a  previous  observation,  however,  brought 
up  to  the  instant  of  the  second  observation,  must  be  trans- 
ferred for  the  exact  run  in  the  interval ;  then  the  intersection 
of  the  two  lines  will  give  the  "  fix  "  at  the  instant  of  second 
observation. 

Intersection  by  Computation-Double  Altitudes. 
Eeferring  to  Fig.  137,  let 

AB  be  the  first  altitude  difference  =  ax , 

CD  be  the  second  altitude  difference  =  a2 , 
Z^  be  the  azimuth  of  the  body  at  the  first  observation, 
Z2  be  the  azimuth  of  the  body  at  the  second  observation. 

Then  the  position  of  A  being  that  by  dead  reckoning  at  the 


644  NAUTICAL  ASTRONOMY 

first  observation,  the  position  of  B  is  obtained  by  running  a 
distance  at  in  the  direction  of  the  first  azimuth,  or  the  oppo- 
site direction,  as  required  by  the  conditions. 

The  position  of  C  (the  D.  R.  position  used  in  the  solution 
of  the  second  observation)  is  obtained  from  the  position  B 
and  the  dead  reckoning  between  observations.  If  desired  the 
run  a±  and  the  run  BC  may  be  combined  in  one  traverse,  thus 
permitting  C  to  be  found  directly  from  A. 

The  position  of  F,  or  fix,  is  obtained  by  running  a  distance 
CF  in  a  direction  parallel  to  that  of  the  first  line  (see,  Ex. 
220  and  Fig.  137).  As  the  line  runs  in  two  opposite  direc- 
tions from  C,  it  is  only  necessary  to  know  that  the  general 
direction  of  CF  is  that  of  CD;  the  direction  of  CF  cannot 
differ  as  much  as  90°  from  that  of  CD,  and  hence  that  direc- 
tion from  the  point  C  is  considered  which  fulfills  this 
requirement. 

The  distance  CF  =  CD  cosec  CFD  =  a2  cosec  (Zt  ~  Z2) 
and  is  easily  found  by  computation,  or  by  using  the  traverse 
tables,  entering  the  tables  with  (Z^^Z2)  as  a  course  and 
taking  out  the  distance  CF  in  the  distance  column  directly 
opposite  the  value  a2  found  in  the  departure  column  (see 
Art.  125). 

The  angle  CFD,  which  equals  the  difference  of  the  azimuths 
of  the  body,  or  bodies,  at  the  two  observations,  or  Z^  ~  Z2 , 
is  always  acute  if  the  same  body  is  observed  in  both  observa- 
tions, and  is  never  greater  than  90°  in  case  the  observations 
are  of  two  different  bodies. 

Intersection  by  Computation-Simultaneous  Observations. 

When  the  position  of  the  ship  does  not  change  between  the 
sights,  and  in  the  case  of  simultaneous  observations,  the  run 
BC  of  Fig.  137  is  zero;  therefore,  in  such  cases,  as  shown  in 


THE  NEW  NAVIGATION  645 

Fig.  138,  when  the  "  fix  "  is  to  be  determined  by  computation, 
use  the  D.  K.  position  A  in  the  solution  of  one  observation, 
apply  the  altitude  difference  a^  to  the  position  A  and  find  the 
computed  point  B  of  the  first  line  BB' ;  then  use  this  com- 
puted point  B  in  the  solution  of  the  other  observation,  apply 
the  altitude  difference  a2  to  the  same  point  B  and  find  the 
computed  point  D  of  the  second  line  DD'.  The  intersection 
F  of  the  two  lines  will  be  the  "  fix  "  (see  Ex.  221). 


FIG.  138. 

Intersection  by  Construction. 

When  the  ship's  position  changes  between  observations,  the 
"  fix "  by  construction  should  be  found  by  first  determining 
the  lines  and  then  plotting  them  on  the  chart  as  indicated 
in  Fig.  137. 

Should  the  ship's  position  not  change  between  sights,  or  in 
the  case  of  simultaneous  observations,  the  computed  point  B 
of  the  first  line  may  be  used  in  the  solution  of  the  second 
sight  and  the  "  fix "  by  construction  found  as  indicated  in 
Fig.  138.  However,  it  might  be  advisable  to  work  both 
sights  by  using  the  co-ordinates  of  one  and  the  same  point  A, 


646 


NAUTICAL  ASTRONOMY 


laying  off  from  that  one  position  both  altitude  differences, 
each  in  its  proper  direction,  for  the  determination  of  the 
computed  points;  then  the  intersection  of  the  position  lines, 


20 


10-- 


50- 


40' 


20' 
-I— 


G.M.T.  12^1  6m 
April  26,  1905 


76  50  W. 


40' 


30' 


20' 


76°10' 


•  -10 


20 


-  -50 


FIG.  139. 


when  drawn,  will  be  the  "fix"  F  (see  Fig.  139).  In  this 
connection  attention  is  called  to  the  method  of  laying  off 
courses  and  distances  on  the  Mercator  and  polyconic  charts 
(see  Art.  31). 


THE  NEW  NAVIGATION 


647 


309.  Special  cases. — The  following  cases  are  specially  re- 
ferred to  in  order  that  the  student  may  learn  how  to  com- 
bine, perhaps  with  advantage  under  certain  circumstances, 
the  direct  methods  of  Chapters  XVII  and  XIX  with  the 
indirect  method  of  Marcq  Saint-Hilaire : 

(1)  When  one  of  the  two  observed  bodies  is  on  the  merid- 
ian (Figs.  140  and  141).  Let  A  be  the  D.  K.  position;  B  the 
computed  point  of  the  line  BB'  (or  that  line  transferred  for 


FIG.  140. 


FIG.  141. 


run  to  the  instant  of  the  second  observation).  Then  whether 
the  observation  of  the  body  on  the  meridian  is  solved  by  the 
Saint-Hilaire  method,  using  the  computed  point  B  of  the 
line  BB'  and  the  altitude  difference  a2 ,  or  by  the  direct 
method  (Art.  240),  the  result  will  be  same;  in  the  one  case 
BD  =  0-2  and  in  the  other  it  is  the  difference  of  latitude  be- 
tween B  and  D.  In  either  case  BF  =  BD  cosec  BFD  = 
BD  cosec  (Z^  ^  Z2)  =  BD  cosec  Z\  ,  but  it  is  unnecessary 
to  find  BF  as  the  latitude  is  well  determined  and  the  longi- 
tude alone  in  doubt.  Find  the  Departure  DF  =  a^  cot  Zt , 
then  the  difference  of  longitude  which  applied  to  the  longi- 
tude of  B  will  give  the  longitude  of  fix. 


648 


NAUTICAL  ASTRONOMY 


In  this  particular  case,  however,  it  must  not  be  forgotten 
that  the  direct  method  of  Arts.  299  and  300,  as  illustrated  in 
Ex.  213,  is  equally  as  simple. 

(2)  When  one  of  two  observations  is  to  be  solved  as  a  time- 
sight.  Many  navigators  are  averse  to  giving  up  the  time-sight 
(tangent)  method  when  conditions  justify  its  use,  at  the  same 
time  preferring  that  of  Saint-Hilaire  to  the  <£"<£'  method. 
This  procedure  may  find  application  in  simultaneous  observa- 
tions of  stars  or  in  forenoon  observations  of  the  sun;  the 

intersection  may  be  found 
graphically  on  the  chart  or  by 
computation  as  indicated  be- 
low and  in  Fig.  142. 

Let  BB'  (Fig.  142)  be  one 
line  from  a  time-sight  of  a 
body  (not  on  the  P.  V.), 
whose  azimuth  is  Z^ ,  trans- 
ferred for  run  to  the  instant 
'^Zj  of  observation  of  another  body 
whose  azimuth  is  Z2 ;  or  a  line 
from  a  time-sight  simultane- 
ous with  the  observation  of  the 
body  whose  azimuth  is  Z2 . 
The  point  B  having  been  used  in  the  solution  of  the  other 
observation  by  the  Saint-Hilaire  method,  D  is  the  computed 
point  of  the  line  DD'  and  F  the  fix.  BFD  —  Z1  ~  Z2 ; 
FBR  =  90°  ~  Z± ;  BE  =  I  and  RF  =  p  between  the  positions 
of  B  and  F.  Entering  the  traverse  tables  with  Zl  ~  Z2  as  a 
course,  look  for  a2  in  the  dep.  column  and  find  BF  in  the  dis- 
tance column;  with  the  direction  of  BF,  that  is  90°  ~  Z^  , 
as  a  course  and  BF  as  a  distance,  take  out  the  corresponding 
I  and  p;  then  find  values  of  Ln  and  D  and,  from  the  co- 
ordinates of  B,  the  latitude  and  longitude  of  "  fix  "  F. 


THE  XEW  NAVIGATION 


649 


(3)  If  the  time  sight  is  of  a  body  observed  on  the  prime 
vertical,  then,  as  indicated  in  Figs.  143  and  144,  the  line  of 
position  BB'  will  run  due  north  and  south,  the  longitude  of 
"  fix "  will  be  well  determined  and  it  will  be  necessary  to 
determine  only  its  latitude.  This  may  be  found  from  the 
latitude  of  B  and  the  difference  of  latitude  between  B  and  F. 
In  this  case  BF  is  this  difference  of  latitude  and  BF  = 
a2  cosec  (Zl  ~  Z2)  =  a2  cosec  (90°  ^  Z2)  =  a2  sec  Z2 . 


zr *-aTH 


FIG.  143. 


FIG.  144. 


However,  attention  is  called  to  the  fact  that  by  the  direct 
methods  the  longitude  would  be  well  determined  from  the 
observation  on  the  prime  vertical  and  that  an  excellent  "  fix  " 
would  then  result  from  using  this  longitude  in  the  solution  of 
the  <£'>'  sight. 

(4)  If  the  Marcq  Saint-Hilaire  method  is  applied  to  the 
first  sight  worked  and  the  latitude  of  the  computed  point  of 
this  first  line  is  used  in  working  a  time-sight  (tangent 
method),  the  intersection,  if  not  found  by  construction,  may 
be  found  as  explained  in  Art.  305. 

The  ship's  most  probable  position. — From  each  one  of  sev- 
eral simultaneous  observations,  a  line  of  position  may  be 


650  NAUTICAL  ASTRONOMY 

obtained  and  from  n  observations  n  lines  will  result;  in  case 
there  have  been  no  errors  of  observations,  or  otherwise,  these 
lines  should  pass  through  one  and  the  same  point.  However, 
there  are  always  errors  which  may  be  due  to  the  imperfection 
of  the  instrument  itself  or  its  adjustment,  to  error  of  the 
tabulated  dip  or  refraction,  to  incorrect  time,  or  the  per- 
sonal equation  of  the  observer,  etc.,  and,  in  consequence,  gen- 
erally speaking,  there  will  be  more  than  one  point  of  inter- 
section, and  there  may  be  for  n  lines  as  many  as 

ft 

points.  It  is  evident  that  the  ship  is  not  at  all  points;  the 
"  theory  of  the  probability  of  errors  "  shows  that  the  most 
probable  position  is  that  point  from  which  if  perpendiculars 
are  drawn  to  the  lines  of  position,  the  sum  of  their  squares 
shall  be  a  minimum.  The  navigator,  in  his  effort  to  check 
a  "  fix  "  from  two  lines  by  means  of  a  third  line  of  position, 
will  often  find  that  the  three  lines  make  a  plane  triangle; 
and,  in  such  cases,  though  the  most  probable  position  may 
easily  be  found  by  construction,  the  practical  navigator,  re- 
garding this  procedure  as  more  a  matter  of  theory  than  of 
practical  value,  will  assume  the  ship's  position  at. the  center 
of  said  triangle,  especially  if  it  is  small  and  equilateral. 

Use  of  Table  44,  Bowditch. — This  table  has  opposite  t  in 
the  p.  m.  column  the  log  sin  \t  in  the  sine  column;  so  if 
using  formula  (239),  look  for  t  expressed  in  time  in  the  p.  m. 
column  and  from  the  sine  column,  directly  abreast,  take  out 
the  log  sin  \t  which,  multiplied  by  2,  will  be  the  log  sin 2  \t. 
This  method  is  illustrated  in  Ex.  221,  and  the  method  of 
considering  the  half-angle  in  degrees  is  illustrated  in  Ex.  220. 


THE  NEW  NAVIGATION 


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656  NAUTICAL  ASTRONOMY 

Ex.  222. — April  7,  1905,  a,  m.,  by  an  observation  of  the 
sun  taken  at  the  G.  M.  T.  Oh  40m  54s  April  8,  1905,  a  ship  was 
faund  to  be  on  a  line  of  position  GM  determined  by  a  posi- 
tion point  G  in  Lat.  39°  27'  N.  and  Long.  69°  18'  W.,  and 
by  the  sun's  true  azimuth  ZN  =  104°  58'.  Expecting  the  ship 
to  maintain  for  several  hours  her  course  320°  (true)  and 
speed  10  knots  per  hour,  the  navigator  decided  to  anticipate 
as  far  as  possible  the  work  for  a  line  from  an  observation  of 
the  sun  to  be  taken  three  hours  later,  or  at  the  G.  M.  T. 
3h  40m  54s.  Assuming  that  the  D.  R.  position  at  that  time 
would  be  Lat.  39°  50'  K  and  Long.  69°  43'  W.,  he  found  for 
that  time  and  place  the  sun's  computed  true  altitude  to  be 
54°  37'  33"  and  its  true  azimuth  or  Z2  to  be  Z*  —  153°  42'. 
The  above  D.  R.  position  and  bearing  were  immediately  laid 
down  on  the  chart  (see  A,  Fig.  145).  The  navigator  failed 
to  get  his  sight  at  the  exact  G.  M.  T.  used  in  the  computation, 
but  16  seconds  later,  or  at  the  G.  M.  T.  3h  41m  10s,  he  observed 
the  sextant  altitude  of  (D  54°  27'  40";  I.  C.  +2';  height  of 
eye  26  feet.  It  is  required  to  find  at  the  time  of  the  second 
observation  a  position  point  and  line  by  the  Marcq  Saint- 
Hilaire  method  and  the  "  fix  "  by  construction. 

Solution. — The  sun's  true  azimuth  at  second  sight  from 
tables,  or  Z2J  is  ZN  =  153°  42',  the  computed  true  altitude 
of  the  sun's  center  is  54°  37'  33",  the  observed  true  altitude 
is  54°  40'  03",  and  hence  &2  is  2'  30"  to  be  laid  off  in  the  direc- 
tion of  Zy  Since  the  G.  M.  T.  of  observation  was  16  seconds 
later  than  that  used  in  the  computation,  the  longitude  of  the 
D.  R.  position  of  the  second  line  must  be  so  changed  as  to 
give  for  the  G.  M.  T.  of  observation  the  same  hour  angle 
as  that  used  in  the  computation;  in  this  case  the  longitude 
must  be  increased  by  16  seconds  of  time,  that  is  by  4  minutes 
of  arc. 

Therefore,  from  A',  which  is  4'  of  longitude  directly  to 
westward  of  A  (see  Fig.  145),  lay  off  a,  =  2'  30",  in  this 


THE 


LITIGATION 


G57 


case,  toward  the  sun  and  B  will  be  a  position  point  of  the 
second  line  BB',  and,  as  AM'  is  the  first  line  brought  up  for 


50 


40 


30 


•9  50  W. 


40' 


30' 


69 W, 


FIG.  145. 


the  run,  ^  in  Lat.  39°  49'  K,  and  Long.  69°  43'  W.,  by  con- 
struction, will  be  the  "  fix."  It  is  apparent  that  a2  could  have 
been  laid  off  from  A  to  0,  and  the  line  then  shifted  to  B  in 
order  to  allow  for  the  error  of  time  (Arts.  31  and  293). 


658 


NAUTICAL  ASTRONOMY 


Littlehales'  graphic  solution. — The  methods  of  the  "  New 
Navigation"  have  been  applied  by  Mr.  G.  W.  Littlehales  of 
the  Hydrographic  Office,  Navy  Department,  to  the  graphic 
solution  of  the  astronomical  triangle  by  means  of  a  stereo- 
graphic  projection  on  the  plane  of  the  observer's  meridian. 
Knowing  the  latitude  and  longitude  by  D.  R.,  and  having 
found  the  observed  body's  hour  angle  and  declination  for  the 


FIG.  146. 

instant  of  observation,  the  navigator  may  find  graphically 
the  values  9f  Z  and  h  for  the  given  D.  R.  position,  and,  from 
these  and  the  measured  true  altitude  of  the  same  body  the 
computed  point  and  line  of  position,  as  previously  explained. 
The  theory  embodied  may  be  briefly  explained  as  follows: 
let  Fig.  146  be  the  projection  referred  to  above  on  a  reduced 
scale;  P,  the  north  pole;  Z,  the  observer's  zenith;  M,  the 
observed  body  located  by  its  hour  angle  ZPM  and  its  decli- 
nation RM ;  then  PZM  is  the  astronomical  triangle  in  which 


THE  NEW  NAVIGATION  659 

the  known  parts  are  PZ  the  co-L,  PM  the  co-d,  and  ZPM  the 
hour  angle  t.  In  this  method  PZ  is  considered  as  <  90° 
or  >  90°  according  as  L  is  north  or  south;  PM  is  <  90° 
or  >  90°  according  as  d  is  north  or  south.  The  required 
parts  of  the  triangle  are  PZM  or  Z  and  ZM  the  co-h,  and 
they  may  be  easily  determined  if  referred  to  a  system  of 
co-ordinates  which,  like  the  equinoctial  system,  admits  of 
permanent  graduations.  This  is  accomplished  by  revolving 
the  astronomical  triangle  about  the  central  point  0  of  the 
projection  with  the  side  PZ  kept  in  coincidence  with  the 
bounding  meridian  till  Z  falls  where  P  originally  was  and  P 
and  M  are  revolved  respectively  into  the  positions  P'  and  M' 
so  that  the  unknown  parts  PZM  and  ZM,  respectively  equal 
to  P'PM'  and  PM',  may  be  measured  from  the  graduations  of 
the  projection ;  PZM  is  reckoned  from  the  left  hand  bounding 
meridian  and  P'PM  from  the  right  hand  bounding  meridian. 
It  is  apparent  that  M  has  described  an  arc  of  a  circle  whose 
radius  is  OM  and  which  subtends  an  angle  equal  to  PZ  or 
co-Zr,  hence  to  obviate  actual  revolution  of  the  triangle,  a 
series  of  equally  spaced  concentric  circumferences  and  a  series 
of  equally  spaced  radial  lines  are  drawn  to  facilitate  identifi- 
cation, the  former  numbered  from  the  center  outward,  the 
latter  numbered  so  as  to  indicate  the  number  of  minutes  of  arc 
estimated  from  08  and  around  to  the  right.  After  having 
plotted  the  body  by  its  hour  angle  and  declination,  it  is  only 
necessary  to  note  the  number  of  the  circumference  and  the 
number  of  the  radial  line  passing  through  the  position  M', 
add  to  this  latter  number  the  distance  ZP  or  co-L  expressed 
in  minutes  of  arc;  the  point  where  the  radial  whose  number 
is  the  latter  sum  intersects  the  noted  circumference  will  be 
the  point  M'  whose  hour  angle  and  declination,  read  from 
the  graduations  of  the  projection,  will  be  respectively  the 
required  Z  and  h  of  the  body  M. 


660  NAUTICAL  ASTROXOMY 

A  stereographic  projection  has  been  constructed  for  a 
sphere  12  feet  in  diameter  which  is  on  such  a  scale  as  to 
admit  of  sufficient  accuracy  for  practical  navigation  and  to 
admit  of  convenient  spacing  of  the  various  meridians,  paral- 
lels, circumferences,  and  radials. 

Each  quadrant  is  subdivided  into  92  overlapping  sections, 
making  868  in  all;  the  plates  representing  them  form  a  book 
of  convenient  size,  each  plate  bearing  the  same  number  as  the 
corresponding  section  of  a  small  projection  called  the  index- 
plate. 

The  point  M  is  roughly  plotted  on  the  index-plate  and  the 
circumference,  radial  line,  and  square  are  each  noted  by  its 
number ;  M'  is  then  roughly  plotted  and  the  square  in  which 
it  falls  is  also  noted. 

Knowing  the  numbers  of  the  squares  and  turning  to  them 
in  the  book  of  plates,  the  positions  of  M  and  M'  are  success- 
ively plotted,  and  the  values  of  Z  and  h  are  taken  from  the 
second  square. 

For  plates  and  further  information  see  Mr.  Littlehales' 
book  "  Altitude,  Azimuth,  and  Geographic  Position." 

Solution  by  nomography. — Lt.  Radler  de  Aquino,  Brazil- 
ian Navy,  has  suggested  a  method  of  finding  li  and  Z  by  using 
a  nomogram  constructed  by  Dr.  Pesci;  this  method,  with 
certain  modifications  introduced  by  the  author,  will  be  found 
explained  in  Appendix  D. 

Solution  by  Tables.* — Navigators  who  find  logarithmic  work 
laborious  may  find  a  position  point  and  a  line  of  position  by 
the  Marcq  St.  Hilaire  method  by  using  tables  from  which 
7i  and  Z  can  be  taken  for  a  position  of  assumed  latitude  and 
longitude,  the  computed  point  being  then  found  by  laying  off 
the  altitude  difference  from  this  assumed  position  in  the  direc- 
tion of  Z  or  180°  -f-  Z,  as  conditions  may  require. 

*  See  "  Altitude  and  Azimuth  Tables,"  by  Lt.  Radler  de  Aquino,  Brazilian  Navy, 
and  "  Altitude  or  Position  Line  Tables,"  by  Frederick  Ball,  R.  N.,  both  books  pub- 
lished by  J.  D.  Potter,  London. 


CHAPTER  XXI. 
DAY'S  WORK  AT  SEA. 

310.  In  the  chapter  on  the  sailings  attention  was  called  to 
the  fact  that  the  general  subject  of  a  day's  work  was  reserved 
till  after  the  student  had  studied  and  understood  the  methods 
of  finding  latitude  and  longitude  by  the  observation  of  celes- 
tial bodies. 

These  methods  having  been  considered,  that  subject  will 
now  be  taken  up. 

In  the  course  of  his  routine  work,  a  navigator,  besides 
determining  his  latitude  at  or  near  noon  and  obtaining  lines 
of  position  from  observations  of  the  sun,  both  a.  m.  and 
p.  m.,  would  get  positions  by  cross  lines  of  stars  or  planets, 
when  conditions  proved  favorable,  as  in  evening  or  morning 
twilight  or  when  moonlight  renders  the  horizon  sufficiently 
distinct.  Polaris,  being  available  in  the  northern  hemisphere, 
should  be  observed  when  conditions  are  favorable,  and  latitude 
desirable. 

The  reckoning  is  estimated  from  the  point  of  departure 
(Art.  123),  or  from  the  noon  position  at  sea  till  noon  of  the 
following  day,  or  till  arrival  at  port  of  destination,  if  the 
voyage  ends  before  noon. 

Owing  to  the  facility  of  getting  the  latitude  at  noon  by 
observations  and  the  fact  that  longitude  can  be  determined 
by  observation  within  a  few  hours  before  noon  and  brought 
up  to  that  time  without  appreciable  error,  it  is  convenient  to 
compare  the  run  by  dead  reckoning  and  by  observation  from 
noon  to  noon,  and  to  regard  the  difference  between  the  noon 


NAUTICAL  ASTRONOMY 


positions  by  dead  reckoning  and  by  observation  as  due  to 
current,  though,  as  a  matter  of  fact,  it  may  be  due  to  other 
causes  as  bad  steering,  faulty  logging,  etc. 

The  navigator  must  report  to  the  commanding  officer  at 
noon  each  day  : 

(1)  Latitude  and  longitude  by  D.  E.  at  noon. 

(2)  Latitude  and  longitude  by  observation  at  noon. 

(3)  Course  and  distance  made  good. 

(4)  Set  and  drift  of  current. 

(5)  The  deviation  of  the  compass  (on  the  course  at  time 
of  a.  m.  sight  perhaps). 

(6)  Course  and  distance  to  destination. 

To  attain  these  results,  the  following  rules  are  laid  down 
for  a  minimum  of  work. 

(1)  Find  the  D.  E-.  positions  at  time  of  a.  m.  sight  for 
longitude  and  at  noon  by  working  the  traverse  from  the  pre- 
vious noon  or  point  of  departure. 

(2)  Find  an  a.  m.  line  of  position  by  either  the  chord  or 
tangent  method  and  the  deviation  of  the  compass  when  the 
sun  is  favorably  situated  for  finding  time.     Plot  this  line  on 
a  Mercator  chart  and  find  graphically  its  intersection  with 
another  line,  if  possible. 

(3)  Find  the  latitude  at  noon  by  observation  from  a  me- 
ridian altitude  of  the  sun,  or  by  reduction  to  the  meridian ;  or 
bring  up  to  noon  a  latitude  obtained  from  the  intersection  of 
a  (j>"cf>'  and  longitude  lines. 

(4)  Take  the  difference  between  the  latitude  by  observation 
and  latitude  by  D.  E.  at  noon,  mark  it  North  or  South  as  the 
former  is  to  the  northward  or  southward  of  the  latter.     This 
discrepancy  may  or  may  not  be  due  to  current,  though  usually 
so  considered  in  the  computation.     Its  value  being  for  24 
hours,  or  from  the  time  of  departure,  a  proportional  part  for 
one  hour,  and  hence  for  the  interval  between  the  forenoon 
sight  and  noon,  may  be  obtained. 


THE  DAY'S  WORK  663 

(5)  Run  the  noon  latitude  back  to  the  time  of  sight,  cor- 
recting backwards  for  both  the  run  from  sight  to  noon  and 
the  proportion  of  current  in  latitude  for  that  time.     The  re- 
sult will  be  the  true  latitude  at  time  of  sight. 

Find  the  longitude  by  observation  at  time  of  sight  by  find- 
ing the  position  point  of  the  line  corresponding  to  the  true 
latitude  at  time  of  sight. 

(6)  The  difference  between  the  longitude  by  observation 
and  by  D.  E.  at  the  time  of  a.  m.  sight  is  a  discrepancy  which 
may  or  may  not  be  due  to  current,  though  usually  so  con- 
sidered in  the  computation.     Its  value  being  from  noon  of  the 
previous  day,  or  time  of  departure,  to  the  time  of  a.  m.  ob- 
servation, a  proportional  part  for  one  hour,  and  hence  for 
the  interval  to  noon,  may  be  found.     It  is  marked  E.  or  W., 
according  as  the  longitude  by  observation  is  to  the  eastward  or 
westward  of  that  by  D.  R. 

(7)  Run  the  longitude  by  observation  at  time  of  sight  up 
to  noon  by  applying  the  run  in  longitude  from  time  of  sight 
to  noon,  and  also  the  current  in  longitude  for  the  same  time, 
each  with  its  proper  sign.     The  result  will  be  the  longitude 
by  observation  at  noon. 

(8)  The  course  and  distance  from  the  noon  position  of  the 
previous  day,  or  point  of  departure,  to  the  noon  position  by 
observation  arrived  at,  will  be  the  course  and  distance  made 
good. 

(9)  The  course  and  distance  from  the  noon  position  ar- 
rived at  by  D.  R.  to  that  by  observation  will  be  the  set  and 
drift  of  the  current,  so-called  (Art.  130). 

(10)  The  course  and  distance  from  the  noon  position  by 
observation  arrived  at  to  point  of  destination  by  middle  lati- 
tude or  Mercator  sailing,  will  be  the  course  and  distance  by 
that  sailing  to  point  of  destination.     Reference  is  made  to 
chapter  VI  for  manner  of  working  dead  reckoning  and  to 
chapters  XVI  and  XVII  for  working  of  sights. 


664 


NAUTICAL  ASTRONOMY 


The  following  problem  will  illustrate  the  points  involved: 
Ex.  223. — On  January  2,  1905,  at  noon,  a  ship's  position 
by  observation  was  latitude  7°  05'  42"  N.,  longitude  148°  19' 
W.  Sailed  thence  until  about  8  a.  m.  next  day  the  following 
courses  and  distances;  wind,  variation,  and  deviation  as  in- 
dicated. 


Wind. 

Course  (p.  c.). 

Var. 

Pev. 

Leeway. 

Distance. 

Sly  and  Wly 
do 
do 
Nly  and  Wly 
do 
do 
do 

301° 
285 
276 
256 
233 
212 
220 

CO  CO  00  00  CO  CO  00 

+++++++ 

-3° 
-4 

-5 
-3 

+1 
+1 

;i° 

6 
3 
3 
3 
0 
0 

22.8  Miles. 
31.6 
34.5 
17.9 
16.1 
It.  9 
12.6 

At  about  8  a.  m.  observed  an  altitude  of  the  sun's  lower 
limb,  23°  42'.  I.  C.  (plus)  1'  20".  Height  of  eye  45  feet. 
Watch  8h  08m  45s.  C— W  10h  03m  15s.  Chronometer  fast  of 
G.  M.  T.  ?m  2P.5.  Sun's  center  bore  (p.  s.  c.)  115°,  ship's 
head  291°,  variation  +  8°.  Work  a  line  of  position,  using 
latitudes  7°  N.  and  7°  20'  N.  Work  an  altitude-azimuth  with 
latitude  7°  20'  X.  and  find  deviation.  (The  azimuth  may  be 
taken  from  tables.) 

Ean  thence  to  noon  291°  (p.  s.  c.),  39  miles,  when  observed 
meridian  altitude  of  sun's  lower  limb,  bearing  South,  59° 
26'  10".  I.  C.  (plus)  1'  20".  Height  of  eye  45  feet. 

1.  Find  latitude  and  longitude  by  D.  E.  at  8  a.  m. 

2.  Work  a  line  for  longitude,  and  find  deviation. 

3.  Find  latitude  and  longitude  by  D.  E.  at  noon ;  true  lati- 
tude at  noon,  and  current  in  latitude;  true  latitude  at  a.  m. 
sight. 

4.  Find  true  longitude  at  8  a.  m.,  current  in  longitude,  and 
true  longitude  at  noon. 

5.  Find  (7N  and  d  made  good,  and  set  and  drift  of  current. 

6.  Find  (7N  and  d  to  Guam  by  Mercator  sailing  using  trigo- 
nometrical formulae. 


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THE  DAY'S  WORK  669 

Had  the  tangent  method  been  used  to  work  the  above  time 
sight,  we  should  have  used  the  latitude  by  D.  E.  at  the  time  of 
a.  m.  sight,  7°  08'  42"  N".,  calling  the  resulting  longitude  com- 
puted longitude. 

With  the  latitude  by  D.  K.,  the  declination,  and  the  L.  A.  T. 
from  the  sight,  the  sun's  true  azimuth,  regarded  as  less  than 
90°,  would  have  been  taken  from  the  azimuth  tables;  and, 
with  this  azimuth  and  the  latitude,  the  value  of  F  found  in 
Table  I.  The  value  of  F  and  the  direction  of  the  line  would 
have  been  written  in  the  form  for  work  thus :  F  =  a  X  . 

Having  found  the  true  latitude  at  the  time  of  a.  m.  sight, 
the  difference  between  it  and  the  latitude  by  D.  E.  at  that 
time  would  have  given  AL,  and,  as  before,  we  should  have  had 
AA  =  AL  X  F. 

Applying  AA  to  the  computed  longitude  at  the  time  of  sight, 
we  should  have  had  the  true  longitude  at  the  time  of  sight. 

The  procedure  from  this  point  would  have  been  the  same  as 
in  the  chord  method  fully  illustrated  in  Ex.  223. 

Marcq  Saint-Hilaire  line. — Had  the  Marcq  Saint-Hilaire 
method  been  used  in  working  the  a.  m.  sight,  we  should  have 
used  the  D.  E.  position  at  the  time  of  sight,  Lat.  7°  08'  42"  K, 
Long.  150°  30'  24"  W.,  in  finding  the  computed  point  of  the 
position  line,  the  arrangement  of  work  in  finding  a  position 
point  of  a  line  as  shown  on  pages  651  and  652  being  substi- 
tuted for  that  on  page  666.  Then  with  the  azimuth  and  the 
latitude  of  the  computed  point,  we  should  have  found  the 
longitude  factor  F  from  Table  I  and  the  direction  of  the  line. 

The  difference  between  the  true  latitude  at  time  of  sight 
and  that  of  the  computed  point  would  have  been  AL  and,  as 
before,  we  should  have  had  AA  =  AL  X  F.  Having  applied 
AA  to  the  longitude  of  the  computed  point  we  should  have 
had  the  true  longitude  at  the  time  of  sight. 

The  procedure  from  this  point  on  would  have  been  the  same 
as  in  the  chord  method  illustrated  in  Ex.  223. 


CHAPTER  XXII. 

TIDAL  WAVES,  TIDAL  CURRENTS, 

AND 
FINDING  TIME  OF  HIGH  WATER. 

311.  Closely  related  to  the  subject  of  the  moon's  meridian 
transit  is  the  subject  of  the  tides,  which,  though  a  very  broad 
one  for  a  work  of  this  scope,  may  be  presented,  even  in  its 
elementary  form,  with  advantage  to  the  student;  by  applying 
general  rules,  he  may  approximate  to  the  time  of  high  water 
for  those  places  not  tabulated  in  the  tide  tables. 

312.  Definitions, — The  phenomena  of  tides,  as  usually  ob- 
served in  tide-water  regions,  are  a  periodic  rise  and  fall  and  a 
recurrent  flood  and  ebb  of  the  water;  the  word  tide  or  tide- 
wave,  properly  refers  to  the  vertical  movement  only,  the  hori- 
zontal movement  being  characterized  as  tidal  current.     The 
maximum  height  to  which  the  tide  rises  is  called  high  water, 
the  lowest  level  to  which  it  falls  low  water;  that  moment  at 
either  high  or  low  water  when  no  vertical  movement  takes 
place  is  called  stand,  and  the  difference  in  height  between  low 
and  high  water  is  called  range. 

Flood  is  the  inflow  of  tide  water  from  the  general  direction 
of  the  ocean,  ebb  its  recession  towards  the  sea;  the  set  of  a 
current  is  the  direction  towards  which  it  is  flowing,  drift  the 
distance  through  which  it  flows  in  a  given  time,  rate  its 
velocity  per  hour,  and  slack  the  term  applied  to  the  period 
between  tidal  currents  when  there  is  no  horizontal  motion. 

313.  Causes  of  the  tides. — The  tides  are  caused  by  the 
difference  of  the  attractions  exerted  by  the  moon,  and,  in  a 


THE  TIDES  671 

less  degree  by  the  sun,  upon  the  earth  and  waters  of  the  earth. 

By  the  law  of  gravitation,  the  attractive  forces  of  the  sun 
and  moon  decrease  as  the  square  of  the  distance  increases,  and 
hence  exert  a  greater  force  on  the  nearer  surface  and  a  less 
force  on  the  farther  surface,  than  on  intermediate  parts;  the 
resultant  effect  being  a  tendency  to  recede  from  the  center  in 
the  parts  not  only  just  under  the  attracting  body,  but  in  the 
parts  diametrically  opposite. 

For  purposes  of  illustration. — The  earth  may  be  considered 
as  surrounded  by  a  uniform  envelope  of  frictionless  water, 
and,  as  illustrated  in  Fig.  147,  let  M  be  the  moon  whose  mean 


FIG.  147. 

attractive  force  on  the  solid  part  of  the  earth  may  be  assumed 
as  acting  at  the  center  E;  therefore,  the  moon  exerts  a  greater 
force  on  the  waters  at  A,  just  beneath  it,  than  on  the  earth 
at  E;  a  greater  force  on  the  earth  at  E,  than  on  the  water  at 
A'  diametrically  opposite.  The  water  at  any  other  position, 
as  at  L,  though  attracted  by  the  moon  less  strongly  than  that 
at  A,  will  have  its  gravity  toward  the  center  diminished,  and 
a  tendency  to  go  toward  A,  due  to  that  component  of  the  force 
along  LM,  which  acts  in  the  direction  of  the  tangent  at  L; 
while  the  water  at  L'  will  have  a  tendency  to  go  toward  Af. 
The  waters  of  the  entire  envelope,  being  free  to  yield  to  a 
similar  tendency,  will  assume  a  spheroidal  shape  with  the 
longer  axis  toward  the  moon,  and  thus  two  tidal  waves,  called 
lunar  waves,  will  be  formed  at  the  points  A  and  A',  These 


672 


NAUTICAL  ASTRONOMY 


will  be  points  of  high  water,  and  midway  between  these  ele- 
vations will  be  depressions  of  the  water  level,  called  low  water, 
as  at  B  and  Bf. 

Number  of  alternations. — Ordinarily  there  are  two  princi- 
pal alternations  of  high  and  low  water  at  a  given  place  in  a 
lunar  day;  and  it  may  be  observed  at  all  places,  except  at 
the  poles  and  on  the  equator,  that  the  two  daily  high  tides 
differ  in  height. 

This  daily  inequality  is  due  to  the  inclination  of  the  plane 
of  the  moon's  orbit  to  that  of  the  equator,  and  to  the  rotation 
of  the  earth  on  its  axis. 


FIG.  148. 

In  Fig.  148,  let  PP'  be  the  earth's  axis,  P  the  North  pole, 
QW  the  plane  of  the  equator ;  let  M  be  the  moon  whose  declina- 
tion is  North,  and  equals  the  angle  MEW;  let  L  be  a  place  on 
the  earth's  surface  having  the  moon  in  its  zenith.  The  tidal 
wave  at  L  is  the  superior  wave,  its  height  may  be  represented 
by  La,  but  at  a  place  L'  in  the  same  latitude,  and  distant  180° 
in  longitude,  the  height  of  the  tide  will  be  represented  by 
L'af;  owing  to  the  revolution  of  the  earth  on  its  axis,  these 
two  places  will  change  situations  with  respect  to  the  moon  in 
about  12  hours,  and  the  height  of  the  tide  at  L  will  then  be 
equal  to  what  it  was  at  L'  12  hours  before.  This  will  be 
known  as  the  inferior  wave  at  L. 

This  alternation  of  high  water  would  theoretically  occur  at 
an  interval  of  12  hours,  if  the  moon . remained  at  rest;  but 


EFFECT  OF  THE  SUN  673 

owing  to  its  advance  to  the  eastward  in  its  orbit,  thereby  arriv- 
ing at  the  same  branch  of  the  same  meridian  later  each  day  by 
a  mean  amount  of  50  minutes,  the  inferior  wave,  or  tide  of  the 
lower  culmination,  will  follow  the  superior  wave,  or  tide  of  the 
upper  culmination,  by  the  average  time  of  12  hours  and  25 
minutes. 

314.  Effect  of  the  sun. — The  attraction  of  the  sun  causes 
in  the  same  way  two  solar  waves  at  diametrically  opposite 
points,  which  reinforce  or  diminish  the  lunar  waves  accord- 
ing to  the  relative  positions  of  the  sun  and  the  moon  in  their 
respective  orbits. 

Owing  to  the  sun's  great  distance,  the  inequality  of  its 
attractions  on  the  earth  and  waters  of  the  earth  is  small, 
and  the  mean  force  of  the  moon  in  causing  tides  is  about  2J 
times  as  great  as  the  sun. 

When  the  sun  and  moon  are  in  conjunction  or  opposition, 
they  act  together  in  producing  the  tidal  wave,  and  the  maxi- 
mum high  and  minimum  low  water  of  the  month  called  spring 
tides  result,  with  maximum  tidal  range;  unusually  high  tides 
would  result  should  the  sun  and  moon  happen  to  be,  respect- 
ively, at  perihelion  and  perigee  at  the  time  of  new  or  full 
moon. 

At  the  first  and  third  quarters  of  the  moon,  the  sun  and 
moon  act  at  right  angles  to  each  other,  and  the  effect  of  the 
solar  wave  is  to  diminish  the  height  of  the  lunar  wave;  the 
minimum  high  and  maximum  low  tides  of  the  month,  called 
neap  tides,  result  with  a  minimum  tidal  range. 

Priming  and  lagging. — When  the  moon  is  in  the  first  and 
third  quarters,  the  solar  wave  is  to  the  westward  of  the  lunar 
wave,  and  there  is  an  acceleration  in  the  time  of  high  water 
called  priming  of  the  tides.  When  the  moon  is  in  the  second 
and  fourth  quarters,  the  solar  wave  is  to  the  eastward  of  the 
lunar  wave,  and  there  is  a  retardation  in  the  time  of  high 
water,  called  lagging  of  the  tides. 


674  NAUTICAL  ASTRONOMY 

315.  Luni-tidal  interval. — The  theoretical  assumptions  in 
the  preceding  article  are  not  fully  justified  by  facts ;  the  earth 
is  not  entirely  covered  with  water,  and  the  water  is  not  fric- 
tionless.     Owing  to  the  rotation  of  the  earth,  the  inertia  of 
the  water,  the  variable  depth  of  the  ocean  bed,  the  obstruc- 
tions offered  by  land,  the  general  contour  of  the  bottom,  and 
the  direction  of  channels,  etc.,  high  tide  is  not  coincident 
with  the  moon's  meridian  transit,  and  the  interval  of  time 
between  the  moon's  meridian  transit  and  the  following  high 
water  is  not  the  same  for  each  day  of  the  month.     These 
intervals  are  known  as  luni-tidal  intervals. 

The  mean  of  these  intervals  on  days  of  new  and  full  moon 
is  called  the  vulgar  or  common  establishment  of  a  port.  It  is 
frequently  spoken  of  as  the  time  of  high  water  on  full  and 
change  days,  being  found  in  the  tidal  data  of  charts  as  H.  W. 
F.  &C. 

The  mean  of  all  the  luni-tidal  high-water  intervals  observed 
throughout  at  least  a  lunar  month,  is  called  the  corrected 
establishment  of  the  port,  and,  when  known,  should  be  used, 
in  preference  to  the  common  establishment,  in  finding  the 
time  of  high  water.  It  will  be  found  tabulated  for  many 
ports  in  Appendix  IV,  Bowditch. 

316.  Age  of  tide. — The  greatest  effect  of  the  sun  and  moon 
in  producing  the  tidal  wave  occurs  at  new  and  full  moon, 
and  the  interval  of  time  from  the  instant  of  new  or  full  moon 
to  the  highest  subsequent  tide  at  any  place  is  known  as  the 
retard  or  age  of  the  tide.     This  varies  with  the  locality,  being 
one  day  on  the  Atlantic  Coast  of  North  America,  and  as  much 
as  2|  days  on  the  Coast  of  England. 

General  laws. — Though  the  subject  of  tidal  waves  is  com- 
plicated by  the  fact  that  the  sun,  moon,  and  earth  do  not 
occupy  the  same  relative  position  more  than  once  in  a  period 
of  about  18  years,  and  by  the  further  fact  that  every  tide  is 
largely  affected  by  local  conditions,  such  as  depth  of  water, 


TIDAL  CURRENTS  C75 

configuration  of  the  coast,  and  even  by  interference  of  differ- 
ent parts  of  the  same  wave ;  still  the  following  elementary  laws 
may  be  laid  down  as  general  for  the  moon's  effect. 

(1)  Two  high  tides  will  occur  daily  at  a  given  place. 

(2)  When  the  declination  of  the  moon  is  0°,  the  two  daily 
tides  at  a  given  place  will  be  equal ;  the  greatest  will  occur  at 
the  equator,  the  least  at  the  poles. 

(3)  When  the  moon's  declination  is  not  0°,  the  two  daily 
tides  at  all  places  except  the  poles  and  equator  will  be  unequal ; 
the  greatest  tides  and  greatest  daily  inequality  will  occur  at 
places  whose  latitude  numerically  equals  the  moon's  declina- 
tion ;  and  the  higher  of  the  two  tides  will  follow  the  moon's 
upper  transit,  when  the  latitude  of  the  place  is  of  the  same 
name  as  the  moon's  declination. 

(4)  The  time  of  high  water  occurs  after  the  moon's  upper 
transit  a  number  of  hours  equal  to  the  establishment  of  the 
port.     The  time  of  the  following  low  water  6  hours  and  13 
minutes  after  high  water,  and  the  time  of  the  next  high  water 
at  a  mean  interval  of  12  hours  and  25  minutes  after  the  first 
high  water. 

Tidal  currents. — A  distinction  must  be  drawn  between  tidal 
waves  and  tidal  currents,  the  former  referring  to  the  vertical 
oscillations  of  the  water,  the  latter  to  the  horizontal  inflow 
and  outflow  caused  by  the  interferences  offered  the  tidal  waves 
by  local  formations  and  the  frictional  resistances  of  the  bot- 
tom and  sides  of  shoal,  narrow  and  contracting  channels,  etc. 

Whilst  it  is  of  importance  to  know  the  times  of  high  water 
when  about  to  enter  or  leave  a  harbor,  it  is  of  more  practical 
importance  in  the  navigation  of  a  vessel  to  be  able  to  antici- 
pate a  probable  set  and  drift  of  a  current  and  to  allow  for 
the  same. 

It  must  not  be  forgotten  that  the  changes  of  tidal  currents 
seldom  correspond  with  high  and  low  waters,  perhaps  never 
except  on  open  coasts  or  in  wide  and  shallow  basins,  certainly 


676  NAUTICAL  ASTRONOMY 

not  in  large  bodies  of  water  having  a  relatively  contracted 
entrance  to  the  sea,  as  in  the  cases  of  Delaware  and  Chesa- 
peake Bays. 

Furthermore,  a  current  in  certain  localities  may  flow  in 
the  offing  one  to  three  hours  after  it  has  turned  along  the 
shore;  such  peculiarities  may  often  be  found  described  in 
the  sailing  directions  of  those  regions  and  should  be  studied 
by  the  navigator. 

In  the  tide  tables  issued  by  the  U.  S.  C.  &  G.  Survey  will 
be  found  current  diagrams  for  Georges  Bank,  Boston  Harbor, 
Nantucket  and  Vineyard  Sound,  New  York  entrance  and 
East  Eiver,  Delaware  and  Chesapeake  Bays,  and  current 
tables,  restricted,  however,  to  points  on  the  Atlantic  and  Pa- 
cific Coasts  of  the  United  States.  In  the  diagrams,  the  set 
and  rate  of  the  current  are  given  for  three  hours  before  and 
after  high  water;  in  the  tables,  for  each  hour  of  the  tide,  at 
some  given  reference  station. 

An  examination  of  these,  when  a  vessel  may  be  in  the  locali- 
ties therein  considered,  will  often  point  out  the  most  favor- 
able conditions  under  which  the  current  should  be  encountered. 

When  lying  in  a  port  of  which  the  tidal  information  is  in- 
complete, and  under  circumstances  that  will  admit  of  obser- 
vations, a  navigator  should  make  every  effort  to  gather  all 
possible  information  about  the  local  currents.  For  the  method 
of  making  tidal  observations  and  a  description  of  the  instru- 
ments used,  etc.,  the  student  is  referred  to  any  standard  work 
on  Marine  Surveying. 

317.  Times  of  high  and  low  water. — The  quickest,  most 
accurate,  and  hence  most  satisfactory  method  of  finding  the 
times  of  high  and  low  water  is  by  taking  this  information 
from  tide  tables,  which  are  furnished  navigating  officers  of 
the  navy.  General  tide  tables  published  by  various  foreign 
governments  may  be  purchased  in  almost  any  seaport;  and 
the  U.  S.  C.  &  G.  Survey  publishes  annually  in  advance  tables 


TIME  OF  HIGH  WATER  677 

containing,  in  addition  to  the  current  matter  referred  to  in 
Art.  316  predictions  as  to  the  times  and  heights  of  every  high 
and  low  water  in  the  following  year  at  certain  principal  ports 
of  the  world  regarded  as  standard  ports  for  tidal  purposes. 
For  these  ports,  the  times  of  tides  are  arranged  in  the  order 
of  the  occurrence  of  tides  in  one  line,  the  corresponding 
heights  above  the  plane  of  reference  (which  for  the  Coast 
Survey  Charts  is  that  of  mean  low  water)  in  a  second  line,  a 
comparison  of  the  heights  indicating  which  are  high  and 
which  are  low  waters.  These  predictions  are  extended  to  over 
1000  other  places  by  applying  to  the  data  of  the  proper  stand- 
ard port,  the  tidal  differences  and  ratios  corresponding  for 
the  places. 

High  water  by  computation. — When  tide  tables  are  not 
available,  the  times  of  high  and  low  water  may  be  found  by 
applying  the  principles  of  rule  4  (Art.  316). 

(1)  Find  the  local  mean  time  of  the  moon's  upper  transit 
at  the  place. 

(2)  Add  to  this  the  high  water  or  low  water  luni-tidal  in- 
terval from  Appendix  IV,  Bowditch,  according  as  the  time  of 
high  water  or  low  water  is  desired.     The  result  will  be  the 
required  time. 

The  H.  W.  luni-tidal  interval,  as  tabulated  in  Bowditch, 
is  the  corrected  establishment  of  the  port;  it  may  be  taken 
from  the  chart  of  the  locality;  or  the  common  establishment 
found  on  the  chart,  as  H.  W.  F.  &  C.,  may  be  used  without 
appreciable  error. 

The  times  given  in  the  Nautical  Almanac  on  page  IV  are 
for  the  astronomical  date. 

When  the  establishment  is  added  to  the  local  time  of  local 
transit,  the  result  will  be  in  astronomical  time;  the  corre- 
sponding civil  time  may  be  a  day  later,  so  if  the  time  of  high 
water  is  desired  for  a  given  civil  date,  and  it  is  found  that 
the  sum  of  the  establishment  plus  the  local  time  of  local  tran- 


678  NAUTICAL  ASTRONOMY 

sit  will  be  greater  than  12  hours,  take  out  the  time  of  transit 
for  the  preceding  date,  since  in  this  case  the  astronomical 
date  is  one  day  less  than  the  civil  date,  and,  when  the  time  is 
converted  into  civil  time,  the  civil  date  of  the  tide  in  question 
will  result. 

Ex.  22Jf. — Find  the  times  of  high  and  low  waters  occurring 
on  January  16,  a.  m.,  1905,  at  Portland,  Me.  Latitude  43° 
39'  28"  N.,  longitude  70°  15'  18"  W. 

In  this  example,  the  sum  of  the  time  of  moon's  transit  and 
the  hmi-tidal  interval  is  greater  than  12  hours;  therefore, 
take  out  the  time  of  transit  for  January  15. 

h    m 

G.  M.  T.  of  Greenwich  transit  Jan.  15,  7  24.9        H.  D.          lm.94 

Corr.  for  Long.  4h.68  W  +     9.08      Long.   W  4h.68 

Corr.     +  9m.08 

L.  M.  T.  of  local  transit  Jan.  15,  7  33.98 

H.  W.  luni-tidal  Int.  Appx.  IV,  Bowditch          11  06 


L.  M.  T.  of  high  water  Jan.  15,  18  39.98 

or  Jan.  16,  6  39.98  a.  m. 

L.  M.  T.  of  local  transit  Jan.  15,  7  33.98 

L.  W.  luni-tidal  Int.  Appx.  IV,      Bowditch          451 


L.  M.  T.  of  low  water  Jan.  15,  12  24.98 

or  Jan.  16,  0  24.98  a.  m. 

Ex.  225. — Find  the  time  of  the  higher  high  tide  that  occurs 
next  after  noon  of  April  9,  1905,  at  Port  Adelaide.  Lati- 
tude 34°  50'  25"  S.,  longitude  138°  26'  58"  E. 

On  April  9,  1905,  the  moon's  declination  is  1ST.;  therefore, 
the  higher  high  tide  occurs  after  the  lower  transit  of  the  moon, 
April  9,  the  time  of  which  may  he  found  as  helow. 

G.  M.  T.  of  Gr.  npper  transit,  April,          9  3  30.7        Mean  H.  D.  2m.06 
G.  M.  T.  of  Gr.  upper  transit,  10  4  20.3         Long.  E        9»»  .23 

2  |     19  7  51  Corr.     —    19»701 

G.  M.  T.  of  Gr.  lower  transit,  April,         9  15  55.5 
Corr.  for  Long.  9»»  23m  E  19.01 

L.  M.  T.  of  local  lower  transit,  April,       9  15  36.49 
H.  W.  Lun.  Int.  Appx.  IV,.  Bowditch  4  04 


L.  M.  T.  of  higher  H.  W.,  April  9,  19  40.49 

or  April  10,  7  40.49     a,  m. 


TIME  OF  HIGH  WATER  679 

Ex.  226. — Find  the  time  of  high  water  occurring  next  after 
noon  on  April  6,  1905,  at  Hong  Kong.  Latitude  22°  16'  23" 
N.,  longitude  114°  10'  02".  E.  Is  this  the  higher  or  lower 
high  tide  of  the  day? 

h     m 

G.  M.  T.  of  Greenwich  transit,  April  6,  1  13.2  H.  D.         lm.81 

Corr.  for  long.  7h. 61  East,  -   13.77         Long.  E   7h  . 61 

Corr.—   13m.77 

L.  M.  T.  of  local  transit,  April  6,  0  59.43 

IT.  W.  luni-tidal  Int.  Appx.  IV,  Bowditch        9  20 


L.  M.  T.  of  high  water,  April  6,  10  19.43 

or  April  6,  10  19.43     p.  m. 

Latitude  and  declination  being  of  the  same  name,  this  is 
the  higher  of  the  two  daily  high,  waters. 


CHAPTEE  XXIII. 
IDENTIFICATION  OF  HEAVENLY  BODIES. 

318.  A  navigator  is  fortunately  not  dependent  on  observa- 
tions of  the  sun  either  in  locating  the  position  of  his  ship  or  in 
determining  the  error   of  his   compass.     Planets   and  fixed 
stars,  Vhen  visible  and  favorably  situated,  are  available  for 
that  purpose.     Owing  to  the  large  number  of  stars  of  the  first 
two  magnitudes  of  differing  right  ascensions,  it  is  probable 
that  several  may  be  found  favorably  situated  for  cross  lines 
at  all  hours  during  twilight,  or  when  the  horizon  may  be  made 
sufficiently  distinct  by  moonlight.     In  these  days  of  fast  ocean 
steamships,  stellar  observations  are  essential  and  an  observer 
with  some  practice  and  a  clear  horizon  should  get  good  re- 
sults from  sights  for  position;  such  sights  should  be  avoided, 
however,  when  the  horizon  is  uncertain.     When  working  for 
compass  error,  it  is  only  necessary  to  see  and  to  know  the  star, 
and    to    obtain    its    compass    bearing,    it   being    immaterial 
whether  the  horizon  is  clear  or  clouded.     The  method  of  ob- 
servation as  well  as  the  methods  of  working  stellar  sights  have 
been  fully  explained. 

319.  Distinction   between   planets   and  fixed   stars. — The 

planets  change  their  positions  in  the  heavens  not  only  with 
reference  to  each  other  but  to  the  fixed  stars;  they  have  a 
perceptible  disc  and  shine  with  a  steady  light;  fixed  stars  do 
not  change  their  positions  relative  to  other  fixed  stars,  and 
they  appear  in  the  most  powerful  glasses  simply  as  luminous 
points  shining  with  a  twinkling  light. 


GROUPING  OF  STARS  CS1 

320.  Distinction  between  planets. — The  only  planets  that 
need  be  considered  by  the  navigator  are  Jupiter,  Venus,  Mars, 
and  Saturn.     Both  Jupiter  and  Venus  are  larger  and  brighter 
than  Sirius ;  when  only  one  is  visible,  it  may  easily  be  taken 
for  the  other,  but  a  comparison  of  the  estimated  right  ascen- 
sion of  the  visible  planet  with  the  tabulated  right  ascensions 
of  Jupiter  and  Venus  will  decide  which  it  is.     When  both 
are  visible,   (1)   the  one  to  the  eastward  will  be  the  one  of 
greater  right  ascension  as  indicated  by  the  tabulated  right 
ascensions  of  the  Nautical  Almanac;  (2)  the  motion  of  Venus 
in  right  ascension  is  more  rapid  than  that  of  Jupiter  and  in 
consequence  its  change  of  position  among  the  fixed  stars  is 
more  noticeable;  (3)  as  Venus  is  an  inferior  planet  with  a 
maximum  elongation  of  about  47°,  it  is  easily  seen  that  as 
morning  or  evening  star,  it  cannot  be  visible  before  sunrise 
or  after  sunset  more  than  three  hours  and  eight  minutes, 
whereas  Jupiter  may  be  visible  at  any  hour  of  night  depending 
on  its  elongation  which,  as  with  all  superior  planets,  varies 
from  0°  to  180°. 

Mars  may  be  recognized  by  looking  up  its  right  ascension 
and  declination ;  it  is  larger  than  a  fixed  star,  and  shines  with 
a  reddish  color,  which  has  caused  it  to  be  known  as  the 
"  Ruddy  Planet." 

Saturn,  owing  to  its  great  distance,  changes  its  relative 
position  among  the  stars  very  slowly,  and  by  the  naked  eye 
may  be  taken  for  a  fixed  star.  Estimating  its  right  ascension, 
or  the  use  of  good  night  glasses,  will  distinguish  it  from  fixed 
stars.  The  three  planets  first  mentioned  are  more  frequently 
used  in  practical  navigation. 

321.  Grouping  and  classification  of  stars. — From  remote 
ages  stars  have  been  grouped  in  constellations,  those  of  each 
constellation,  as  a  rule,  being  arranged  in  order  of  brightness 
and  distinguished  by  having  Greek  or  Roman  letters  prefixed 
to  the  name  of  the  constellation,  or  by  numerals  when  both 


G82  NAUTICAL  ASTRONOMY 

alphabets  have  been  exhausted,  the  brightest  star  of  the  group 
being  represented  by  the  letter  a.  Specific  names  are  usually 
given  to  the  most  conspicuous  stars. 

Stars  are  found  in  nautical  almanacs,  arranged  according 
to  their  right  ascensions  and  classified  by  magnitudes  or 
brightness,  the  lowest  magnitude  assigned  to  stars  just  visible 
to  the  naked  eye  being  the  sixth.  Assigning  to  sixth  magni- 
tude stars  an  average  brightness  of  unity,  and  regarding  the 
stars  of  one  magnitude  about  2J  times  as  bright  as  those  of 
the  lower  magnitude,  the  average  brightness  of  first  magnitude 
stars  should  be  100.  .  Of  course,  there  are  marked  deviations 
from  this  rule,  the  most  notable  exception  being  Sirius,  which 
is  perhaps  500  times  as  bright  as  a  star  of  the  sixth  magnitude. 

322.  Navigational  stars.  —  The  twenty  brightest  stars: 
a  Canis  Majoris  (Sirius),  a  Argus  (Canopus),  a  Aurigae  (Ca- 
pella),  a  Bootis  (Arcturus),  a2  Centauri,  a  Lyrse  (Vega), 
(3  Orionis  (Eigel),  a  Eridani  (Achernar),  a  Canis  Minoris 
(Procyon),  {3  Centauri,  a  Aquilae  (Altair),  a  Crucis,  a  Orionis 
(Betelgeux),  a  Tauri  (Aldebaran),  a  Virginis  (Spica), 
a  Scorpii  (Antares),  ft  Geminorum  (Pollux),  a  Piscis  Aus- 
tralis  (Fomalhaut),  a  Leonis  (Regulus),  a  Cygni  (Deneb), 
and  perhaps  a  dozen  more  may  be  classed  as  navigational 
stars,  and  every  navigator  should  be  able  to  recognize  these 
and  to  select  the  ones  most  favorably  situated  for  his  purposes. 
To  do  so,  it  is  useless  to  make  a  study  of  the  constellations 
based  on  the  fanciful  grouping  of  stars  by  the  ancients;  it  is 
only  necessary  to  know  (1)  one  conspicuous  constellation  in 
the  northern  heavens  about  which  to  group  stars  of  North 
declination;  (2)  one  in  the  region  of  the  equinoctial  leading 
to  a  knowledge  of -others  in  the  same  region,  to  some  one  of 
which,  stars  of  either  North  or  South  declination  up  to  certain 
limits  may  be  referred;  (3)  one  in  the  southern  hemisphere 
that  may  assist  in  locating  the  stars  adjacent  to  the  South 
celestial  pole. 


POINTS  OF  REFERENCE  G83 

323.  Constellations  of  reference. — The  constellations  rec- 
ommended for  obvious  reasons  in  carrying  out  the  above  plan 
are  (1)  Ursa  Major  or  "The  Dipper"  ;  (2)  Orion;  (3)  the 
Southern  Cross.     The  student  having  made  himself  familiar 
with  the  visible   stars   of   these   constellations,   and  having 
learned  certain  bright  stars  near  them,  should  trace  out  others 
in  one  of  three  ways : 

(1)  by  bearings  and  angular  distances; 

'(2)  by  prolonging  a  line  (straight  or  curved)  passing 
through  two  known  stars  till  at  a  certain  approximate  dis- 
tance it  may  pass  through  a  required  star; 

(3)  by  the  geometrical  figures,  which  in  many  cases,  three 
or  more  bright  stars  form  with  each  other. 

The  first  method  is  unsatisfactory  as  the  bearing  of  one 
star  from  another  is  a  great  circle  bearing  and  should  be  noted 
when  the  known  star  is  at  its  upper  culmination  and  as  near 
the  zenith  as  possible — conditions  seldom  governing.  An  in- 
spection of  star  maps  on  the  Mercator  projection  would  only 
confuse  the  student  as  the  bearings  there  shown  are  not  great 
circle  bearings. 

The  second  and  third  methods  in  connection  with  Plates 
VI  to  IX  will  perhaps  be  found  the  best  and  most  expeditious 
methods  for  indentifying  stars  when  the  surrounding  heavens 
are  visible. 

324.  Description  of  Plate  VI. — The  plate  shows  the  princi- 
pal stars  in  the  northern  hemisphere  whose  declination  ex- 
ceeds 30°.     The  Eoman  numerals  on  the  margin  show  the 
meridians  of  right  ascension  at  intervals  of  one  hour.     As  the 
right  ascension  of  the  meridian  is  the  L.  S.  T.,  if  the  observer 
faces  the  North  and  holds  the  plate  so  that  the  numeral  which 
represents  the  L.  S.  T.  at  the  time  of  observation  is  upper- 
most, the  stars^  in  the  upper  part  of  the  plate  will  be  shown 
in  the  same  relative  positions  as  they  appear  in  the  heavens. 

If  the  observer  faces  the  North  and  holds  the  plate  so  that 


PLATE  VI. 


1 

T\ 
^  •                             \\ 

$   •  • 

^           .                                                                                                      X 

Polaris'^ 
O  Aurigae 
or  Capella 

A     J^ 

ffLyrae^f 
or  Vega*-«            / 

THE  PRINCIPAL  STARS  AROUND  THE  NORTH  CELESTIAL  POLE  or  A 
DECLINATION  GREATER  THAN  30°  N. 


STARS  IN  PLATE  YI  685 

the  name  of  a  month  found  in  the  margin  is  uppermost,  the 
plate  will  show  the  visible  heavens  around  the  pole  as  they  ap- 
pear about  8.30  p.  m.  in  that  particular  month;  the  number  of 
stars  in  the  lower  part  of  plate  cut  off  by  the  horizon  depend- 
ing on  the  latitude  of  the  observer. 

Ursa  Major,  commonly  called  the  "  Dipper  "  from  its  shape, 
one  of  the  brightest  and  most  conspicuous  of  the  northern 
constellations,  consists  of  seven  principal  stars.  Beginning 
with  the  edge  of  the  bowl  they  are  (a)  Dubhe,  (ft)  Merak, 
(y)  Megrez,  (8)  Phecda,  (e)  Alioth,  (£)  Mizar,  (^  Benet- 
nasch.  The  first  two  (a  and  ft)  are  the  brightest  and,  point- 
ing to  the  pole  star  (Polaris),  are  known  as  the  pointers. 
Polaris  is  the  principal  star  of  Ursa  Minor  which  appears  also 
in  the  shape  of  a  smaller  dipper,  Polaris  being  in  the  extremity 
of  the  handle. 

Cassiopeia. — About  the  same  distance  from  Polaris  as  the 
"Dipper,"  but  on  the  opposite  side,  is  Cassiopeia's  chair, 
whose  five  principal  stars  appear  in  the  form  of  the  letter  M 
or  W,  according  to  the  position  of  the  constellation  in  its 
diurnal  path. 

ft  Cassiopeia. — A  line  from  y  Ursse  Majoris  through  Polaris, 
produced  about  30°,  leads  to  (3  Cassiopeia. 

a  Cassiopeia. — A  line  from  8  Ursse  Majoris  through  Polaris 
leads  to  a  Cassiopeia  called  Schedir,  the  farthest  one  of  the 
chair  from  the  pole  star. 

Square  of  Pegasus. — A  line  from  the  pointers  through  Po- 
laris, produced  beyond  Cassiopeia,  leads  first  to  ft  Pegasi 
(Scheat),  then  to  a  Pegasi  (Markab),  two  stars  in  a  notice- 
able figure  resembling  a  square ;  the  other  two  being  y  Pegasi 
(Algenib)  and  a  Andromedse  (Alpheratz),  the  latter  nearer 
the  pole  (Plates  YI  and  YIII). 

a  Lyrse  or  Vega. — A  line  from  y  passing  between  8  and  « 
Ursse  Majoris  leads  to  Yega,  a  very  bright  star  of  a  decided 


686  NAUTICAL  ASTRONOMY 

blue  tint,  which  is  attended  by  five  other  stars,  making,  with 
Vega,  two  triangles. 

a  Cygni  or  Deneb. — A  line  from  y  through  8  Ursae  Majoris 
extended  passes  between  Vega  and  Deneb.  Also  a  line  from 
Algenib  through  Scheat  (Plate  VIII),  continued  to  nearly 
twice  its  distance,  leads  to  Deneb. 

a  Aquilae  or  Altair. — A  line  from  Polaris  midway  between 
Vega  and  Deneb  leads  to  Altair,  which  is  further  distinguished 
by  having  an  attendant  star  each  side  of  it  and  by  proximity 
to  the  Dolphin  which  shows  five  stars,  four  of  which  form  a 
small  diamond  (Plate  VIII).  Altair,  Vega,  and  Deneb 
form  a  triangle  nearly  right  angled  at  Vega  (Plate  VIII). 

a  Aurigse  or  Capella. — A  line  from  y  Ursa?  Majoris  passing 
between  the  pointers  (a  and  ft  Ursse  Majoris)  leads  to  Capella, 
a  very  bright  star  of  a  yellow  tinge,  attended  by  a  small  tri- 
angle of  three  stars  to  the  southward  of  it  called  "  the  kids." 

A  line  from  the  middle  star  of  Orion's  belt  through  Orion's 
head  and  ft  Tauri  leads  to  Capella  (Plate  VII). 

Capella,  Algol,  and  Aldebaran  form  an  equilateral  triangle 
(Plate  VII). 

a  Bootis  or  Arcturus. — A  line  from  Dubhe  passing  between 
y  and  8  Ursse  Majoris  leads  to  Arcturus,  and  the  handle  of 
the  dipper  curves  toward  it. 

Arcturus  is  a  very  bright  star  with  a  reddish  tint,  is  at- 
tended by  a  small  triangle  of  three  stars,  is  as  far  from  the 
pointers  on  one  side  as  Capella  is  on  the  opposite  side;  it 
forms  bold  triangles  with  Spica  and  Eegulus,  also  with  Spica 
and  Antares,  both  triangles  nearly  right  angled  at  Spica 
(Plate  VIII). 

a  and  ft  Geminorum — Castor  and  Pollux. — A  line  from  8 
Ursae  Majoris  passing  between  the  pointers  leads  to  Castor 
and  Pollux,  which  are  about  as  much  one  side  of  the  Dipper  as 
the  Northern  Crown  is  the  other  side.  A  line  from  the 


STARS  IN  PLATE  VII  687 

middle  star  of  Orion's  belt  (Plate  VII)  through  Betelgeux 
leads  to  Castor,  which  shines  with  a  greenish  light.  Betel- 
geux, Procyon,  and  Pollux  (Plate  VII)  form  a  triangle, 
right  angled  at  Procyon. 

a  Leonis  or  Regulus. — A  line  from  8  Ursse  Majoris  passing 
between  pi  and  y  Ursse  Majoris  leads  to  Eegulus.  This  is  a 
bright  white  star  and,  being  in  the  handle  of  the  so-called 
sickle  or  reaping  hook,  is  a  very  prominent  one.  It  forms  a 
triangle  with  Spica  and  Arcturus,  right  angled  at  Spica 
(Plate  VIII). 

325.  Description  of  Plates  VII  and  VIII.— The  principal 
stars  of  a  declination  less  than  45°,  North  and  South,  are 
shown  in  these  plates,  those  whose  right  ascensions  are  be- 
tween 0  and  XII  hours  in  Plate  VII,  those  of  a  right  ascen- 
sion greater  than  XII  hours  in  Plate  VIII. 

If  about  8.30  p.  m.  in  a  particular  month,  these  plates  be 
so  held  overhead  that  the  feathered  arrow  points  North  whilst 
the  Eoman  numerals  increase  to  the  eastward,  then  the  bright- 
est stars  of  the  heavens  near  the  meridian  will  be  those  stars 
in  the  plates  whose  right  ascensions  are*  indicated  by  figures 
below  the  name  of  the  given  month. 

326.  Orion  and  the  stars  it  leads  to. — Orion,  the  most  beau- 
tiful constellation  of  the  heavens,  consists  of  a  quadrilateral 
formed  of  three  bright  stars  and  one  of  lesser  magnitude,  the 
figure  being  longer  in  the  North  and  South  direction.     The 
NE.  star  is  a  Orionis  (Betelgeux) ;  the  NW.,  y  Orionis  (Bel- 
latrix) ;  and  the  SW.,  /?  Orionis  (Rigel). 

Within  the  quadrilateral  are  three  small  stars,  nearly  equi- 
distant, and  in  a  line  nearly  NW.  and  SE.,  forming  what  is 
known  as  Orion's  belt.  Nearly  midway  between  the  two 
northern  stars  and  a  little  further  to  the  northward  are  three 
small  stars  forming  a  triangle  in  the  imaginary  head  of  Orion. 

a  Canis  Majoris  or  Sirius. — This,  the  brightest  star  of  the 
heavens,  shines  with  a  scintillating  white  light.  The  three 


PLATE  VII, 


a  I 

§  I 

2  I 


•a 


PLATE  VIII, 


690  NAUTICAL  ASTRONOMY 

stars  of  Orion's  belt  point  southeastward  to  Sirius,  which 
forms  an  equilateral  triangle  with  Betelgeux  and  Procyon. 
Sirius,  Rigel,  and  the  triangle  in  Orion's  head  form  a  triangle 
right  angled  at  Rigel. 

a  Canis  Minoris  or  Procyon. — A  line  from  Bellatrix  through 
Betelgeux,  curving  to  the  southward  and  eastward,  leads  to 
Procyon,  a  star  of  a  yellowish  tint.  A  line  from  Arcturus 
through  Denebola  and  Regulus  leads  to  Procyon. 

a  Tauri  or  Aldebaran. — A  line  from  Betelgeux  through  the 
three  stars  in  Orion's  head  and  extended  to  three  times  the 
distance  leads  to  Aldebaran,  which,  shining  with  a  decided 
reddish  tint,  is  conspicuous  as  forming  a  V  with  four  other 
stars. 

a  Arietis  or  Hamel. — A  line  drawn  from  Betelgeux  through 
Aldebaran  leads  to  Hamel,  which  may  be  known  by  two  small 
stars  southwestward  of  it.  Hamel,  Menkar,  and  a  Tauri  form 
a  triangle  nearly  right  angled  at  Menkar. 

/?  Leonis  or  Denebola. — A  line  from  Procyon  through  Regu- 
lus leads  to  Denebola  at  a  little  over  half  the  distance.  For 
Regulus,  see  Art.  324. 

a  Virginia  or  Spica. — About  35°  SE.  from  Denebola  is 
Spica,  a  bright  white  star,  which  forms  with  Arcturus  and 
Denebola  an  equilateral  triangle.  Four  stars  of  the  constel- 
lation Corvus  form  the  shape  of  a  "  spanker,"  the  gaff  point- 
ing to  Spica. 

a  Scorpii  or  Antares. — A  line  from  the  Dolphin  through 
Altair  leads  to  Antares  which  is  a  bright  star  of  a  decided 
reddish  tinge,  forming  with  adjacent  stars  the  approximate 
figure  of  a  hand  glass,  Antares  at  junction  of  glass  and 
handle.  It  forms  with  a  and  (3  Librae  a  long  triangle, 
a  Libras  being  on  a  line  between  Spica  and  Antares.  A  line 
from  Regulus  through  Spica  extended  to  the  same  distance 
passes  a  little  to  the  southward  of  Antares.  A  line  from 
a2  Crucis  through  (3  Centauri,  produced  three  times  its  length, 
leads  to  Antares  (Plate  TX). 


PUTE  IX. 


PRINCIPAL  STARS  AROUND  THE  SOUTH  CELESTIAL  POLE  OT  A 
DECLINATION  GREATER  THAN  30°  S. 


G92  XAUTICAL  ASTRONOMY 

a  Piscis  Australis  or  Fomalhaut. — A  line  from  Scheat 
through  Markab  extended  about  45°  leads  to  Fomalhaut, 
which  forms  with  three  other  stars  an  irregular  quadrilateral. 
It  forms  an  equilateral  triangle  with  a  Pavonis  and  Achernar 
(Plate  IX). 

The  Pleiades. — A  line  from  midway  between  Algenib  and 
Alpheratz  through  a  Arietis,  extended  the  same  distance, 
leads  to  the  Pleiades,,  a  remarkable  cluster,  of  which  six  stars 
are  visible  to  the  naked  eye. 

327.  Description  of  Plate  IX. — This  plate  shows  the  prin- 
cipal stars  of  the  southern  hemisphere  whose  declination  ex- 
ceeds 30°.     What  was  said  in  Art.   324  about  the  Eoman 
numerals  and  najnes  of  months  around  the  margin  of  Plate 
VI,  apply  to  the  numerals  and  months  of  this  plate  with  this 
exception,  that  the  observer  faces  the  South. 

The  Southern  Cross. — This  is  the  most  conspicuous  constel- 
lation of  the  southern  hemisphere,  and  is  outlined  by  four 
bright  stars;  when  the  cross  is  above  the  pole,  a2  Crucis  is 
the  southernmost,  /?  Crucis  the  easternmost,  y  Crucis  the  north- 
ernmost, and  8  Crucis  the  westernmost  star  of  the  cross.  As 
a  line  through  a  and  /?  Centauri  points  directly  to  the  cross, 
those  two  stars  are  known  as  the  pointers. 

a  Argus  or  Canopus. — This  star  is  next  to  Sirius  in  bril- 
liancy, is  midway  betwen  Eigel  and  the  cross.  Rigel,  a  Co- 
lumbae,  Canopus,  J3  Argus,  and  a2  Crucis  are  at  equal  dis- 
tances apart  in  a  very  slightly  curved  line. 

a  Eridani  or  Achernar. — Is  about  midway  between  Canopus 
and  Fomalhaut,  forms  an  equilateral  triangle  with  a  Pavonis 
and  Fomalhaut,  also  with  {3  Argus  and  a  Columbse. 

328.  In  cloudy  weather. — In  case  the  surrounding  heavens 
are  clouded  and  it  is  desired  to  ascertain  the  name  of  a  single 
star  that  may  be  out,  its  altitude  and  azimuth  having  been 
taken,  the  name  may  be  found  in  the  Nautical  Almanac, 
from    its    right   ascension    and    declination    obtained    thus: 

NOTE. — Having  a  star's  observed  h  and  Z  the  G.  M.  T.  of  observation,  and 
observer's  position,  t  and  d  may  be  found  from  Aquino's  "  Altitude  and  Azimuth 
Tables."  The  L.  S.  T.  and  t  will  give  the  R.  A.  which,  with  the  declination,  will 
identify  the  star. 


FINDING  THE  COORDINATES  693 

(1)  when  the  body  is  on  the  meridian,  its  right  ascension  is 
the  L.  S.  T.  of  the  instant  of  observation  (Art.  173),  and  the 
declination  may  be  found  from  the  known  latitude  of  the  place 
and  the  measured  true  meridian  altitude;  (2)  in  case  the  body 
is  not  on  the  meridian,  the  approximate  right  ascension  (the 
L.  S.  T.  at  transit)  may  be  obtained  from  the  local  time  of 
observation  and  the  body's  estimated  hour  angle,  and  the 
approximate  declination  from  the  known  latitude  and  the 
estimated  altitude  at  transit;  (3)  the  star  may  be  projected 
stereographically  from  its  observed  altitude  and  azimuth,  and 
the  right  ascension  and  declination  determined  with  sufficient 
accuracy  to  distinguish  its  name;  (4)  the  coordinates  may  be 
obtained  by  the  use  of  a  celestial  globe;  (5)  having  deter- 
mined by  observation  the  true  altitude  and  azimuth  of  a 
heavenly  body,  its  right  ascension  may  be  found  by  applying 
to  the  L.  S.  T.  of  observation  the  hour  angle  taken  from  the 
azimuth  tables,  as  explained  in  Appendix  C,  and  the  declina- 
tion may  then  be  found  from  the  formula 

cos  d  =  sin  Z  cos  h  cosec  t* 

This  formula  will  give  the  numerical  value  of  the  declination 
but  will  not  determine  the  sign,  about  which,  however,  there 
should  be  no  ambiguity  except  when  the  declination  is  very 
small.  Having  determined  the  numerical  value  of  the  decli- 
nation by  computation,  enter  the  tables  with  L,  t  and  -\-d 
(that  is  of  same  name  as  latitude)  and  see  if  the  azimuth 
found  tabulated  there  agrees  with  that  used  in  the  computa- 
tion. If  agreement  is  found,  the  declination  is  properly 
marked ;  if  not,  the  declination  is  negative  and  verification 
should  be  sought  on  that  supposition.  With  the  values  of 
L,  t  and  Z,  an  inspection  of  the  tables  in  most  cases  will 
determine  the  value  of  the  declination  without  the  necessity 
of  computation. 

When  inspecting  the  Nautical  Almanac  or  a  star  table  in  an 
effort  to  identify  an  observed  heavenly  body,  through  an  agree- 
ment of  tabulated  coordinates  with  those  obtained  by  any  of 
the  methods  referred  to  above,  the  navigator  should  always 
consider  the  possibility  of  having  observed  a  planet  instead  of 
a  fixed  star. 

*The  use  of  this  formula  in  this  connection  was  first  proposed  by 
Lt.-Coradr.  G.  W.  Logan,  U.  S.  N.,  in  The  Proceedings  of  U.  S.  Naval 
Institute,  No.  104. 


694  NAUTICAL  ASTRONOMY 

There  are  various  graphic  methods  for  determining  the 
names  of  stars;  those  proposed  by  Admiral  Sigsbee  and  Lt.- 
Comdr.  Rust,  IT.  S.  N.,  Mr.  G.  W.  Littlehales  of  the  Hydro- 
graphic  Office,  and  Lt.  Radler  de  Aquino  of  the  Brazilian 
Navy,  are  among  the  best.  For  the  details  of  Admiral  Sigs- 
bee's  method  the  student  is  referred  to  H.  0.  chart  No.  1560 ; 
for  Comdr.  Rust's  method  to  The  Proceedings  of  IT.  S.  Naval 
Institute  Nos.  116,  123,  and  124;  for  Mr.  Littlehales'  method 
to  his  admirable  work  "  Altitude,  Azimuth,  and  Geographical 
Position";  for  Lt.  Radler  de  Aquino's  method  to  the  nomo- 
gram  explained  in  Appendix  D. 

It  is  well,  when  navigating,  to  note  at  twilight  the  appoxi- 
mate  bearings  and  altitudes  of  prominent  stars  whose  names 
are  known,  whether  desired  for  observations  or  not;  then 
under  circumstances  above  referred  to,  a  single  bright  star 
peeping  out  from  the  clouds  at  a  time  when  an  observation  is 
desired  might  be  recognized  from  its  approximate  position, 
noted  about  the  same  time  a  night  or  two  before,  when  the 
weather  conditions  were  such  as  to  make  identification  without 
question. 

Ex.  227. — At  sea,  January  19,  1905,  a.  m.,  in  latitude  by 
D.  R.  50°  33'  N.  and  longitude  by  D.  R.  40°  04'  W.,  weather 
cloudy,  a  bright  star  was  observed,  through  a  break  in  the 
clouds,  on  the  meridian  bearing  South ;  star's  sextant  altitude 
51°  54'  10";  I.  C.  +3';  height  of  eye  36  feet;  W.  T.  of  obser- 
vation 2h  llm  10s;  C— W  2h  39m'55s;  chronometer  slow  on 
G.  M.  T.  lm  10s ;  what  was  the  name  of  the  star  ? 

h  m  a  o  I  n  in 

W.  T.                     2  11  10  *18  \  51  54  10  S  .  I.  C.  +  3    00 

C-W                       2  39  56  Corr.  -  3  38  Dip  -  5    53 

C.C.                   +  1  10  ^  H  „  88  „  Ret.  .-_OJ5 

SAJ't  16    53    16  *'sz          88   09    88  N  Corr.    -  8   38 

RA.M.0          194863.64  Latitude   60    83    00  N 

Corr.  G.  M.  T.    +      2    46.287  *'sd  12    23    32  N 


G.  S.  T.  12    43    54.927 

Long.  W.  2    40    16 

10    03    38.927 


An  inspection  of  the  mean  place  tables  of  the  Nautical 
Almanac  of  1905  shows  the  above  star  to  have  been  a  Leonis 
(Regulus)  whose  tabulated  coordinates  were:  R.  A.  10b  03m 


FINDING  THE  COORDINATES  695 

18S.838,  dec.  N.  12°  25'  54'.'22.  In  practice,  at  sea,  it  will 
be  unnecessary  to  correct  the  R.  A.  M.  O.  for  the  G.  M.  T., 
an  approximate  R.  A.  being  generally  sufficient  for  the  identi- 
fication of  the  star. 

Ex.  228.—  April  5,  1905,  about  7h  llm  18s,.  p.  m.,  of  local 
mean  time,  in  latitude  by  D.  R.  20°  40'  S.  and  longitude  by 
D.  R.  90°  12'  E.,  weather  cloudy,  observed  a  bright  star 
through  a  break  in  the  clouds;  star's  sextant  altitude  25°  58' 
40";  I.  C.  +1';  height  of  eye  19  feet;  star's  bearing  (p.  s.  c.) 
or  ZNz=307°,  variation  —  8°,  ship  heading  East  (p.  s.  c.), 
deviation  -j-2°.  Required  the  hour  angle  and  name  of  star  ? 

h     m       s  o       /      11  in 

L.  M.  T.  7    11    18  *'s  ha         25    58    40  I.  C.          +1    00 

Long.  E.  6    00    48  Corr.  -      5    15  Dip  -  4    16 


R.M.o  06228.24 

Corr.  G.  M.  T.  11.581 

G.S.T.  2    03    09.821  *'•**  <P-  «.  c.)  80T° 

Long.  E.  6    00    48 


4    30    21.821 


*'sZN(true)  301 

jfc'sZ  (true)  S121W 

^8  Z  (expressed  in  time)  = 


Entering  the  azimuth  tables  (H.  0.  Pub.  No.  120)  in 
the  given  latitude  20°  40'  S.,  with  the  altitude  25°.9  in  the 
declination  column,  and  Z  (expressed  as  time)  8h  04m  in  the 
hour  angle  column,  we  find  tabulated  in  the  usual  place  of  the 
azimuth  the  star's  hour  angle  t  —  53°  24'  W.  or  +  3h  33m  36s. 
Converting  the  L.  M.  T.  of  observation  into  L.  S.  T.  and 
applying  the  hour  angle  to  this  L.  S.  T.,  the  star's  right  ascen- 
sion is  found  to  be  4h  30m  22s;  then,  by  substitution  in  the 
formula  cos  d  —  sin  Z  cos  h  cosec  tf  we  find  d  =  16°  09'. 

This  is  evidently  North  for,  by 

Z=121°  sin        9.93307  •          .1          h       "    r    aT1(q    /    wuv, 

h  =  25°  53'  25"     co8        9.95406     ^     vg    V  •          v?     *v     -IPO    An/ 

t  =  53°24'  cosec  10.  09538  declination  North  16  09,  we 
d=  16°  09'  cos  9.98251  will  obtain  by  inspection  of  the 

tables  the  proper  azimuth  which 

would  not  be  obtained  on  the  supposition  that  the  declination 
is  South.  An  inspection  of  the  Nautical  Almanac  of  1905 
will  show  this  star  to  have  been  a  Tauri  (Aldebaran),  its 
coordinates  in  the  mean  place  tables  being:  R.  A.,  4b  30m 
288.083;  and  d,  N.  16°  19'  07?31. 


CHAPTER  XXIY. 
GENERAL  OBSERVATIONS. 

329.  Watchfulness  over  the  compasses. — The  navigator 
should  see  the  compasses,,  especially  the  "  standard/'  properly 
centered  in  their  binnacles  and  so  swung  that  the  rims  of  the 
bowls  will  be  horizontal;  that  the  cards  traverse  freely  in 
the  bowls  and  that  the  bowls  swing  properly  in  their  gimbals ; 
that  the  lubber's  lines  are  in  the  fore-and-aft  line  of  the  ship ; 
that  each  compass  has  sufficient  sensibility,  magnetic  mo- 
ment, and  steadiness;  that  there  is  no  concealed  iron  in,  or 
near,  the  binnacle  that  may  probably  disturb  the  compass; 
that  the  electric  light  wires  in  the  vicinity  of  the  compass  are 
double,  the  direct  and  return  wires  close  together  and  well 
insulated,  and  that  there  is  no  short  circuit  through  the 
bridge  or  binnacle  fittings;  that  quartermasters  and  others 
near  the  compass  have  no  steel  grommets,  side  arms,  bayonets, 
nor  other  metal  on  their  persons  that  may  affect  the  compass ; 
that  all  correctors  are  in  place  and  the  quadrantal  correctors 
are  free  from  magnetism. 

When  entirely  free  from  magnetism,  these  spheres  will  have 
a  magnetic  axis  in  the  direction  of  the  line  of  force;  North 
or  red  polarity  being  to  northward  of  and  below  the  center, 
South  or  blue  polarity  to  southward  of  and  above  the  center, 
however  rapidly  they  may  be  rotated  or  their  positions 
changed.  The  point  of  North  polarity  should  attract  the 
South  end  of  a  magnetic  needle  and  repel  the  North  end;  if 
it  does  not,  the  particular  sphere  possesses  magnetism  which 
should  be  removed  by  heating  it  to  a  dark  heat  only,  covering 
with  ashes,  and  allowing  it  to  slowly  cool. 


COMPASSES  697 

After  compensating,  care  must  be  taken  to  see  each  magnet 
system  secured  in  its  proper  direction,  the  clamp  screw  being 
set  up  when  the  lines  traced  on  the  frame  and  supporting 
plate  of  the  carrier  are  in  coincidence;  also,  that  the  key  is 
removed  from  the  mechanism  of  both  carriers  when  they  are 
at  the  proper  height. 

Attention  is  also  called  to  the  fact  that  the  quadrantal  cor- 
rectors may  be  accidentally  moved.,  even  when  the  securing 
nuts  are  well  set  up. 

When  steaming  under  forced  draft,  all  compasses,  especially 
one  near  the  smoke  stack,  should  be  carefully  watched;  there 
are  cases  on  record  of  marked  changes  in  compass  deviations, 
due  to  changes  in  the  magnetic  character  of  the  casing  and 
smoke  stack,  caused  by  the  intense  heat  of  the  escaping  gases 
under  such  circumstances. 

Unceasing  vigilance  the  only  safe-guard. — The  fact  cannot 
be  too  strongly  impressed  on  the  inexperienced  navigator  that 
however  well  a  compass  may  be  compensated  at  one  moment, 
this  compensation  cannot  be  assumed  as  correct  at  a  later 
time,  whether  in  the  same  or  a  different  locality,  unless  there 
is  a  frequent  determination  or  verification  of  the  deviations. 
It  has  been  shown  that  the  deviations  will  change  on  change 
of  magnetic  latitude,  or  of  longitude,  if  it  involves  a  change  of 
magnetic  dip  or  horizontal  force,  and  from  other  causes ;  and 
the  navigator  must  anticipate  these  changes  and  ascertain  the 
deviations  of  his  compasses  daily,  in  port  and  at  sea,  on  as 
many  points  as  possible,  at  times  when  the  dynamos  are  in 
operation  under  (1)  normal,  (2)  full  loa'd. 

Record  of  compass  observations. — A  systematic  record 
should  be  kept  of  all  observations  for  compass  deviations  in 
the  Compass  Journal  provided  for  that  purpose. 

Special  use  of  the  standard. — A  ship's  course  should  al- 
ways be  directed  by  the  standard  compass,  and  all  courses 
and  bearings  entered  in  the  ship's  log-book  should  be  those 


698  NAUTICAL  ASTRONOMY 

shown  by  that  compass  alone ;  since  all  bearings,  before  being 
used,  must  be  corrected  by  the  deviation  due  to  the  direction 
of  the  ship's  head  at  the  time  they  were  taken,  it  is  apparent 
that  the  ship's  heading  (p.  s.  c.)  should  also  be  recorded 
whenever  a  bearing  (p.  s.  c.)  is  entered  in  the  log-book. 

Comparison  of  compasses. — Whenever  a  ship  is  steadied  on 
a  course  (p.  s.  c.),  the  headings  by  the  steering,  pilot  house, 
and  check  compasses  should  be  noted  and  recorded  in  the 
rough  note-book,  so  that  if,  from  any  cause,  the  standard 
becomes  deranged,  the  fact  may  be  made  immediately  appar- 
ent. It  would  be  a  useful  practice  at  sea  to  record  as  above 
the  headings  by  all  compasses  at  "  eight  bells."  The  officer- 
of-the-deck,  during  his  watch,  should  frequently  compare  the 
readings  (of  the  standard  and  steering  compasses,  especially 
when  the  ship  heels ;  and,  whenever  the  course  is  changed,  he 
should  personally  see  the  ship  on  her  proper  course  per  stand- 
ard and  that  both  the  quartermaster  and  helmsman  have  noted 
the  corresponding  heading  per  steering  compass. 

Change  in  position  of  correctors. — Whilst  it  is  desirable  to 
keep  the  deviations  of  the  standard  a  minimum,  still  no  radi- 
cal change  should  be  made  in  the  position  of  correctors  ex- 
cept when  it  is  evident  a  permanent  change  of  deviations  has 
taken  place  and  in  cases  where  an  opportunity  may  be  given 
to  obtain  residual  deviations  on  at  least  16  points  before 
proceeding  to  sea,  and  then  only  by  permission  of  the  cap- 
tain, as  required  by  the  Naval  Eegulations. 

Effects  of  lightning. — Marked  changes  in  compass  devia- 
tions in  iron  or  steel  ships  may  occur  if  the  ship  is  struck 
by  lightning,  even  when  the  condition  of  ship  and  fittings  is 
otherwise  unaffected.  A  case  is  of  record  in  which  the  mag- 
netic polarity  of  a  ship  and,  in  consequence,  the  sign  of  its 
compass  deviations  were  entirely  reversed  by  a  stroke  of  light- 
ning. 

Effects  of  target  practice. — The  magnetism  of  a  ship  and 


THE  "GAUSSIN"  ERROR  699 

hence  its  compass  deviations  are  liable  to  change  during 
target  practice,  especially  when  heavy  guns  are  fired  and  the 
ship  is  kept  on  one  heading  during  the  practice. 

These  effects  may  be  minimized  if  the  practice  takes  place 
with  the  ship's  head  in  all  quadrants ;  or,  at  least,  if  half  the 
practice  occurs  with  the  ship's  head  in  one  direction,  and  the 
other  half  with  the  ship's  head  in  the  opposite  direction. 

Immediately  after  the  practice,  the  ship  should  be  swung 
for  deviations,  though  these  may  be  expected  to  change,  as 
the  effects  of  the  firing  will  dissipate  after  a  few  days.  A 
record  should  be  kept  of  the  changes,  and  the  ship  again 
swung  at  the  first  opportunity  after  the  ship's  magnetism  has 
become  normal. 

It  is  reasonable  to  expect  a  less  change  in  the  deviations, 
due  to  target  practice,  in  an  old  ship  than  in  one  recently 
launched. 

Effect  of  retentive  magnetism  or  "  The  Gaussin  Error." — 
When  a  ship  is  kept  continuously  on  or  near  one  compass 
heading  for  several  days,  whether  at  anchor,  alongside  a  dock, 
or  underway,  and  especially  if  underway  and  subjected  to 
much  vibratory  motion,  a  temporary  magnetic  character  is 
produced  in  the  "soft"  iron  and  iron  of  qualities  interme- 
diate between  "  soft "  and  "  hard." 

The  time  a  ship  is  on  a  particular  heading  plays  an  im- 
portant part  in  the  intensity  of  the  magnetism  induced,  and 
affects  the  rapidity  of  its  disappearance  on  change  of  the 
ship's  direction;  the  effects  of  retentive  magnetism  are  par- 
ticularly noticeable  in  vessels  on  a  northerly  or  southerly 
course  after  having  steered  for  some  time  on  an  easterly  or 
westerly  course,  and  more  so  in  high  than  in  low  latitudes. 

Having  headed  westerly  for  several  days,  if  the  course  is 
changed  to  the  southward,  a  deviation  of  +  sign  will  be  in- 
troduced for  the  new  course;  if  the  course  is  changed  to  the 
northward,  a  deviation  of  ( — )  sign  will  be  introduced;  all  in 


700  NAUTICAL  ASTRONOMY 

addition  to  the  tabulated  value  for  that  heading  (p.  c.).  The 
reverse  happens  for  a  vessel  which  has  been  steaming  for  any 
length  of  time  on  easterly  courses. 

When  changing  course  in  the  vicinity  of  dangers,  the  navi- 
gator must  remember  this  tendency  of  the  compass  needle  to 
deviate  in  the  direction  from  which  the  course  was  changed, 
and,  if  possible,  check  the  deviations  by  observations  of  the 
sun,  planets,  or  stars ;  if  the  weather  proves  thick  or  foggy,  he 
must  proceed  with  extreme  caution,  lay  his  course  with  a  large 
allowance  for  safety,  and  on  soundings  "  use  the  machine  " 
or  "  keep  the  lead  going." 

330.  Sextant. — This  subject  has  been  exhaustively  treated 
in  Chapter  IX,  and  it  only  remains  now  as  a  matter  of 
emphasis  to  again  caution  the  navigator  to  always  handle  his 
sextant  with  care,  to  guard  it  from  all  jars  or  shocks,  and  un- 
necessary exposure  to  the  sun's  rays,  to  heat,  or  dampness. 
It  should  be  put  away  before  target  practice  and  not  left,  as 
valuable  sextants  have  been  left,  on  a  chart  table  in  the  pilot 
house  to  be  blown  off  by  the  blast  of  a  bow  gun. 

After  the  instrument  has  been  once  properly  adjusted,  there 
should  be  no  reason  for  its  getting  out  of  adjustment,  if  care- 
fully handled;  however,  this  adjustment  must  be  frequently 
verified  and  the  I.  C.  must  be  obtained  every  time  an  obser- 
vation is  taken. 

The  careful  navigator  will  not  only  not  lend  his  sextant  to 
any  one,  but  he  will  permit  nobody  but  himself  to  handle  it. 

331.  Chronometers. — The  navigator  should  bear  in  mind 
the  fact  that  chronometers  are  delicate  instruments,  worthy 
of  all  the  care  and  attention  recommended  for  them  in  Chap- 
ter X.    Their  error  and  rate  should  be  frequently  determined 
and  always  just  before  sailing. 

When  in  cruising  grounds,   away   from  any  place  where 


CHRONOMETERS  701 

chronometers  may  be  rated,  the  errors  of  all  chronometers 
should  be  checked  up  occasionally  from  the  second  differences, 
which  may  be  done  provided  no  two  of  the  chronometers  are 
running  alike,  that  is,  have  rates  of  the  same  amount  and 
sign,  and  provided  further  that  the  temperature  curve,  as 
determined  on  board,  of  one  of  the  chronometers  will  give  a 
close  approximation  to  the  rate  of  that  chronometer  for  the 
mean  temperature  of  the  elapsed  time ;  each  chronometer  may 
be  tried  for  this  purpose  in  the  effort  to  satisfy  the  equations 
below. 

The  method  embodies  simply  the  solution  .of  two  equations 
with  three  unknown  quantities,  the  value  of  one  being  as- 
sumed from  the  best  data  obtainable  as  a  trial  value  in  the 
determination  of  the  others. 

Let  x,  y,  and  z  be  the  mean  daily  rates  of  chronometers 
A,  B,  and  C,  respectively,  since  last  rating. 

Let  a  be  the  mean  value  of  the  second  differences  of  A 

and  B, 

b,  the   mean   value   of   the   second   differences   of   A 
and  C; 

then  the  equations  will  be  of  the  general  form, 

x-\-y  —  a, 
x  +  z  —  ~b. 

Having  found  x,  y,  and  z,  the  G.  M.  T.  corresponding  to  a 
given  absolute  instant  of  time  as  found  from  each  chronom- 
eter should  be  the  same. 

It  must  not  be  forgotten  that  x,  y,  and  z  are  the  mean  rates 
for  the  interval  since  previous  rating,  and  each  may  differ 
from  the  real  rate  at  the  instant  considered. 

332.  Caution  as  to  charts. — As  required  by  naval  regula- 
tions, the  navigator  must  familiarize  himself  with  the  sailing 


702  NAUTICAL  ASTRONOMY 

directions  and  charts  of  his  cruising  grounds,  see  that  they  are 
of  recent  date,  and  give  the  latest  obtainable  information. 

When  shaping  a  course  through  localities  where  the  varia- 
tion may  have  a  considerable  change  in  an  ordinary  day's 
run,  notably  off  the  Newfoundland  and  North  American 
coasts  and  in  the  westward  approaches  to  the  English  Chan- 
nel, it  would  be  advisable  to  measure  off  roughly  the  prob- 
able day's  run  and  note  the  change  of  variation  in  that  time. 
If  the  number  of  degrees  of  change  is  say  6°  W.  in  24  hours, 
it  is  plain  that  a  magnetic  course  laid  down  at  the  beginning 
of  the  day  cannot  be  continued  throughout  it  without  involv- 
ing a  large  error,  in  position ;  therefore,  it  is  advisable  to  alter 
the  course  1°  at  regular  intervals,  and  to  allow  for,  say,  an 
increase  of  6  °  westerly  variation,  proceed  thus : 

Having  found  at  the  outset  the  correct  true  course,  obtain 
the  first  compass  course  by  applying  to  the  true  course  the 
proper  deviation  and  the  mean  variation  from  the  chart  for 
the  first  four  hours'  run;  at  the  end  of  this  and  each  suc- 
ceeding four  hours'  run  of  the  day,  alter  the  course  by  1°,  and 
to  the  right  since  the  variation  is  westerly. 

However  carefully  the  work  may  be  done,  absolute  accuracy 
in  a  position  plotted  by  run  or  cross  bearings  cannot  be  ex- 
pected. The  paper  used  is  dampened  in  the  making  of  the 
chart  and  distortion  takes  place  on  drying;  this  distortion 
varies  with  the  quality  and  size  of  the  paper.  Besides,  charts 
will  become  distorted  from  use  in  damp  or  foggy  weather. 
For  these  reasons,  the  navigator  should  not  rely  entirely  on 
positions  by  graphic  methods,  and  those  entered  in  the  log- 
book should  be  by  computation. 

In  the  selection  of  charts  for  use  in  navigating,  those  of 
the  largest  scale  available  should  be  selected;  they  will  surely 
possess  greater  detail,  and  perhaps  may  be  from  plates  on 
which,  in  view  of  the  scale  of  the  chart,  the  latest  corrections 
have  been  made. 


CAUTION  AS  TO  CHARTS  703 

Close  attention  should  be  given  to  the  date  of  the  survey 
from  which  a  chart  resulted,  especially  if  the  chart  is  of 
regions  where  the  bottom  is  of  shifting  sand.  For  instance, 
on  Nantucket  shoals  the  bottom  is  of  such  a  shifting  nature 
that  charts  of  that  locality  of  only  a  recent  date  should  be 
regarded  as  reliable. 

Considering  the  fact  that  these  shoals  extend  miles  beyond 
the  points  at  which,  when  surveying,  signals  may  be  carried, 
it  is  evident  that  many  soundings,  even  in  shoal  water,  have 
been  located  by  D.  E.  alone. 

Soundings  so  located,  in  such  regions,  where  currents  are 
strong  and  variable,  should  be  regarded  with  suspicion,  and 
the  prudent  navigator  will  always  navigate  those  shoals  with 
caution. 

Close  attention  should  also  be  given  to  the  amount  of  detail 
on  the  charts  used ;  when  charts  show  soundings  to  be  few  and 
far  between  in  the  vicinity  of  shallow  water  or  of  an  occasional 
reef,  it  is  wise  to  presume  that  the  locality  has  been  only  par- 
tially surveyed,  and  that  perhaps  there  are  many  dangers 
near,  even  if  uncharted. 

If  the  soundings,  even  though  few,  show  only  deep  water, 
it  is  fair  to  assume  that  the  intervening  spaces,  in  which  no 
soundings  are  given,  may  be  navigated  with  safety. 

333.  Before  going  to  sea,  or  entering  pilot  waters. — Be- 
fore reaching  pilot  waters,  the  navigator  should  see  all  charts 
of  the  locality  corrected  to  date;  should  study  these  charts, 
the  sailing  directions,  and  the  light  and  buoy  lists  of  those 
waters;  and  should  see  the  sounding  machine  in  good  order 
and  ready  for  use. 

Before  going  to  sea,  he  should  see  the  log  lines  properly 
marked,  sand  glasses  timed,  patent  logs  in  good  order,  and 
should  have  obtained  the  deviations  of  the  standard  compass, 
the  error  and  rate  of  all  chronometers  on  board.  Before  en- 
tering or  leaving  port,  he  should  personally  see  the  lead  lines 


704  NAUTICAL  ASTRONOMY 

well  soaked  in  water,  stretched,  and  properly  marked.  To 
facilitate  the  work  of  marking  the  log  and  lead  lines,  the 
required  distances  should  be  laid  off  in  a  suitable  place  on 
deck,  or  on  the  fore-and-aft  bridge,  and  permanently  marked 
with  copper  tacks. 

334.  Discrepancy  in  a.  m.  and  p.  m.  sights. — Abnormal 
refraction  may  be  looked  for  in  the  Eed  Sea,  Persian  Gulf, 
and  in  the  regions  of  the  Gulf  stream,  or  wherever  there  is  a 
marked  difference  between  the  temperature  of  the  air  and  the 
water. 

When  such  refraction  exists,  there  will  be  an  apparent  dis- 
crepancy between  the  results  obtained  from  forenoon  and 
afternoon  sights,  because  the  error  introduced  by  using  the 
tabulated  dip  affects  them  the  opposite  way;  therefore,  under 
such  circumstances,  it  would  be  better  to  reduce  the  longi- 
tude from  both  a.  m.  and  p.  m.  sights  to  noon,  and  then  to 
take  the  mean  of  these  two  resulting  longitudes  as  the  correct 
noon  longitude. 

Fortunately,  however,  a  navigator,  when  navigating  the 
aforementioned  waters,  is  not  restricted  in  taking  sights  to 
times  when  refraction  is  abnormal ;  star  sights  may  most  likely 
be  obtained  at  more  favorable  times. 

335.  Error  of  a  ship's  position. — It  must  not  be  forgotten 
that  a  ship's  position  at  sea  is  only  approximate,  even  when 
determined  by  the  most  exact  navigator;  under  the  most  favor- 
able circumstances,  and  with  the  best  instruments  obtainable. 

The  extent  of  error  depends  on  instrumental  imperfections, 
errors  of  tabulated  dip  and  refraction,  error  of  time,  errors  of 
observation,  and,  in  the  case  of  double  altitudes,  the  error  of 
the  intervening  run;  these  may  be  increased  by  the  circum- 
stances of  unfavorable  location  of  the  body  observed,  of  bad 
weather,  or  rough  seas. 


ERROR  OF  POSITION  705 

Under  average  conditions,  the  ship's  position  may  be  as- 
sumed in  doubt  at  least  two  miles,  and  as  the  sign  of  this  prob- 
able error  may  be  +  or  ( — )>  "the  ship's  position  may  be  any- 
where within  a  circle  described  from  the  determined  position 
as  a  center  with  a  radius  of  2  miles,  and  courses  should  be 
shaped  with  this  uncertainty  in  view. 

When  desiring  to  lay  a  course,  from  a  position  determined 
at  sea,  to  pass  a  danger,  it  would  be  prudent  to  multiply  the 
assumed  average  error  by  a  number  (2,  3,  or  4,  according 
to  circumstances),  called  a  "coefficient  of  safety,"  and,  con- 
sidering the  result  obtained  as  the  limit  of  possible  error,  to 
describe  a  circle  about  the  determined  position  with  that 
limit  as  a  radius;  then  to  shape  the  course  from  a  point  on 
that  side  of  the  circle  nearest  the  danger  to  be  passed. 

The  general  principle  embodied  in  the  use  of  a  "  coefficient 
of  safety  "  was  more  correctly  applied  in  connection  with  Sum- 
ner  lines  as  explained  in  Arts.  293  and  294,  wherein  a  ship's 
position  was  shown  to  be  somewhere  within  a  parallelogram 
formed  by  drawing  parallels  on  each  side  of  each  line  of  posi- 
tion and  at  such  distances  as  to  include  errors  of  altitude, 
time,  etc.,  which  might  be  assumed  by  the  navigator  as  prob- 
able under  existing  conditions.  By  using  the  parallelogram, 
the  navigator  is  better  enabled  to  see  in  which  direction  his 
position  is  the  most  in  doubt  (see  Figs.  126,  127,  128). 

336.  The  advisability  of  keeping  landmarks  in  sight. — 
Considering  the  uncertainty  of  positions  at  sea,  it  is  prudent, 
when  navigating  coasts  well  charted,  lighted,  and  buoyed, 
especially  when  there  are  outlying  lightships,  as  in  the  case 
of  the  eastern  coast  of  the  United  States,  to  make  certain 
landmarks  or  lights  in  regular  succession  and  at  short  inter- 
vals of  time,  being  careful,  however,  to  see  that  the  ship  is  not 
set  by  currents  into  regions  of  possible  danger. 

If  by  so  doing  a  vessel  is  not  taken  too  much  out  of  the 
direct  course  to  destination,  it  is  always  advisable,  whenever 


706  NAUTICAL  ASTRONOMY 

possible,  to  sight  a  mark  from  which  to  take  a  fresh  departure, 
whether  making  ready  to  close  in  with  the  land  for  the  pur- 
pose of  entering  port,  or  to  put  out  to  sea  in  view  of  approach- 
ing fog  or  bad  weather. 

In  pilot  waters,  the  ship's  position  should  be  located  by 
observations  of  permanent  landmarks  if  practicable,  as  buoys 
are  frequently  out  of  place  and  lightships  are  sometimes  so, 
even  when  just  replaced,  on  the  station.  In  the  vicinity  of 
dangers,  whether  buoys  are  in  sight  or  not,  the  use  of  the 
danger  angle  is  advisable  (see  Arts.  118  and  119). 

In  running  a  channel,  where  there  are  no  well-defined  land- 
marks, and  as  to  which  there  may  be  some  doubt,  the  ship 
may  be  steered  through  it  by  zig-zagging  occasionally  from 
side  to  side,  keeping  the  lead  going,  and  thus  showing  on 
which  side  of  the  channel  the  vessel  may  be  and  in  which 
direction  the  course  must  be  changed  to  find  deeper  water, 
care  being  taken,  however,  not  to  run  into  dangerously  shoal 
water. 

In  going  in  or  out  of  port,  try  to  pick  up  a  range,  ahead 
or  astern  as  the  case  may  be;  and,  as  local  currents  may  be 
uncertain,  watch  for  any  possible  indications  of  their  set  and 
strength,  such  as  the  riding  of  buoys,  the  general  heading  of 
vessels  at  anchor,  or  the  opening  of  the  range  on  which  the 
ship  may  be  steering.  A  comparison  of  the  courses  and  dis- 
tances sailed  by  compass  and  those  made  good  as  indicated  by 
bearings  will  give  the  set  and  drift  of  the  current. 

Before  anchoring,  the  navigator  should  know  not  only  the 
set  of  the  tide  but  the  maximum  rise  and  fall  to  ensure 
having,  at  low  tide,  sufficient  water  under  the  bottom.  When 
desiring  to  find  an  anchorage  on  two  bearings,  approach  it 
upon  that  bearing  which  may  be  the  most  convenient  one, 
reduce  speed,  and  stop  in  sufficient  time  to  let  go  the  anchor 
when  the  ship  is  also  upon  the  second  bearing. 


CAUTION  AS  TO  DATA  707 

337.  Disregarding  the  seconds  of  data  when  solving  the 
astronomical  triangle. — The  ship's  position  being  subject  to 
the  errors  enumerated  in  Art.  335,  and  hence  uncertain,  even 
under  the  most  favorable  conditions,  some  navigators  believe 
themselves  justified  in  using  their  data  only  to  the  nearest 
minute  of  arc  in  solving  the  astronomical  triangle.  If  the 
errors  due  to  such  procedure  were  known  to  offset  other  errors, 
such  theories  would  be  tenable;  but  as  it  is  equally  probable 
that  they  would  augment  them,  it  seems  advisable,  in  the 
absence  of  any  definite  knowledge  as  to  the  effect  of  neglect- 
ing the  seconds,  to  exercise  great  care  in  obtaining  data  and 
then  to  use  the  values  obtained,  when  working  sights  for 
either  latitude  or  longitude. 


ros 


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TABLE  IV. 

USED  FOB  CALCULATION  OF  COEFFICIENTS  B,  C,  D,  AND  E. 
PRODUCTS  OF  ARCS  MULTIPLIED  BY  THE  SINES  OF  15°  RHUMBS. 


ARCS. 

sa 

Sin.  15° 

S2 

Sin.  30" 

S3 

Sin.  45* 

S4 

Sin.  60° 

S5 

Sin.  75° 

ARCS. 

o     / 

O           1 

O          1 

o       / 

o      / 

O          f 

o     / 

0    0 

0    0 

0    0 

0    0 

0    0 

0    0 

0    0 

010 

0    3 

0    5 

0    7 

0    9 

010 

010 

020 

0    5 

010 

014 

017 

019 

020 

030 

0    8 

015 

021 

026 

029 

030 

040 

010 

020 

028 

035 

039 

040 

050 

013 

025 

035 

043 

.  048 

050 

1    0 

016 

030 

042 

*052 

058 

1    0 

1  10 

018 

035 

049 

1    1 

1    8 

1  10 

1  20 

021 

040 

057 

1    9 

117 

1  20 

130 

023 

045 

1    4 

118 

127 

1  30 

1  40 

026 

050 

111 

127 

137 

1  40 

1  50 

028 

055 

118 

135 

146 

1  50 

2    0 

031 

1    0 

125 

144 

156 

2    0 

210 

034 

1    5 

132 

153 

2    6 

210 

220 

036 

110 

139 

2    1 

215 

220 

230 

039 

115 

146 

210 

225 

230 

240 

041 

120 

153 

219 

235 

240 

250 

044 

125 

2    0 

227 

244 

250 

3    0 

047 

130 

2    7 

236 

254 

3    0 

310 

049 

135 

214 

245 

304 

310 

320 

052 

140 

221 

253 

313 

320 

330 

054 

145 

229 

3    2 

323 

330 

340 

057 

150 

236 

311 

333 

340 

350 

1    0 

155 

243 

319 

342 

350 

4    0 

1    2 

2    0 

250 

328 

352 

4    0 

410 

1    5 

2    5 

257 

337 

4    1 

410 

420 

1    7 

210 

3    4 

345 

4  11 

420 

430 

110 

215 

311 

354 

421 

430 

440 

112 

220 

318 

4    2 

430 

440 

450 

115 

225 

325 

411 

4  40 

450 

PRODUCTS  or  ARCS  BY  SINES  721 

TABLE  IV. — Products  of  Arcs 
Multiplied  by  the  Sines  of  15°  Rhumbs. —  (continued). 


ARCS. 

Si 

Sin.  15° 

S2 

Sin.  30° 

S3 

Sin.  45° 

S4 

Sin.  60° 

S5 

Sin.  75° 

ARCS. 

0        / 

0           f 

0           , 

0           / 

0           / 

O           1 

o     / 

5    0 

118 

230 

332 

420 

450 

5    0 

510 

120 

235 

339 

428 

459 

510 

520 

1  23 

240 

346 

437 

5    9 

520 

530 

125 

245 

353 

446 

519 

530 

540 

128 

2  50 

4    0 

4  54 

528 

540 

550 

131 

255 

4    7 

5    3 

538 

550 

6    0 

133 

3    0 

4  15 

512 

548 

6    0 

610 

136 

3    5 

422 

520 

557 

610 

620 

138 

310 

429 

529 

6    7 

620 

630 

141 

315 

436 

538 

617 

630 

640 

1  44 

320 

443 

546 

626 

640 

650 

146 

325 

450 

555 

636 

650 

7    0 

149 

330 

457 

6    4 

646 

7    6 

710 

151 

335 

5    4 

612 

655 

710 

720 

154 

340 

511 

621 

7    5 

720 

730 

1  56 

345 

518 

630 

715 

730 

740 

159 

350 

525 

638 

724 

740 

750 

2    2 

355 

532 

647 

734 

750 

8    0 

2    4 

4    0 

539 

656 

744 

8    0 

810 

2    7 

4    5 

546 

7    4 

753 

8  10 

820 

2    9 

410 

554 

713 

8    3 

820 

830 

212 

415 

6    1 

722 

813 

830 

840 

215 

420 

6    8 

730 

822 

840 

850 

217 

425 

615 

739 

832 

850 

9    0 

220 

430 

622 

748 

842 

9    0 

910 

222 

435 

629 

756 

851 

910 

920 

225 

440 

636 

8    5 

9    1 

920 

930 

228 

445 

643 

814 

911 

930 

940 

230 

450 

650 

822 

920 

940 

950 

233 

455 

657 

831 

930 

950 

722  NAUTICAL  ASTRONOMY 

TABLE  IV. — Products  of  Arcs 
Multiplied  by  the  Sines  of  15°  Rhumbs. — (continued). 


ARCS. 

B, 

Sin.  15° 

S2 

Sin.  30° 

S3 

Sin.  45° 

S4" 
Sin.  60° 

S5 

Sin.  75° 

ARCS. 

0        / 

o       / 

0           / 

o       / 

0           1 

0           t 

O        1 

10    0 

235 

5    0 

7    4 

840 

940 

10    0 

1010 

238 

5    5 

711 

848 

949 

1010 

1020 

240 

510 

718 

857 

959 

1020 

1030 

243 

515 

725 

9    6 

10    9 

1030 

1040 

246 

520 

733 

914 

1018 

1040 

10  5C 

248 

525 

740 

923 

1028 

1050 

11    0 

251 

530 

747 

932 

1038 

11    0 

11  10 

253 

535 

754 

940 

1047 

11  10 

11  20 

256 

540 

8    1 

949 

1057 

11  20 

1130 

259 

545 

8    8 

958 

11    6 

11  30 

11  40 

3    1 

550 

815 

10    6 

1116 

11  40 

1150 

3    4 

555 

822 

1015 

1126 

11  50 

12    0 

3    6 

6    0 

829 

1024 

1135 

12    0 

1210 

3    9 

6    5 

836 

1032 

1145 

1210 

1220 

312 

610 

843 

1041 

1155 

1220 

1230 

314 

615 

850 

1050 

12    4 

1230 

1240 

317 

620 

857 

1058 

1214 

1240 

1250 

319 

625 

9    4 

11    7 

1224 

1250 

13    0 

322 

630 

912 

1116 

1233 

13    0 

1310 

324 

635 

919 

11  24 

1243 

13  10 

1320 

327 

6  40 

926 

1133 

1253 

1320 

1330 

330 

645 

933 

1141 

13    2 

1330 

1340 

332 

650 

940 

1150 

1312 

1340 

1350 

335 

655 

947 

1159 

1322 

1350 

14    0 

337 

7    0 

954 

12    7 

1331 

14    0 

1410 

340 

7    5 

10    1 

1216 

1341 

1410 

1420 

343 

710 

10    8 

1225 

1351 

1420 

1430 

345 

715 

10  15 

1233 

14    0 

1430 

1440 

348 

7  20 

1022 

1242 

1410 

1440 

1450 

350 

725 

1029 

1251 

1420 

1450 

PRODUCTS  OF  ARCS  BY  SINES 

TABLE  IV. — Products  of  Arcs 
Multiplied  by  the  Sines  of  15°  Rhumbs. —  (continued). 


723 


ARCS. 

Si 

Sin.  15' 

S2 

Sin.  30° 

S3 

Sin.  45° 

S4 

Sin.  60° 

S5 

Sin.  75° 

ARCS. 

0        / 

O          f 

o       / 

o       r 

o      / 

o      / 

0       / 

15    0 

353 

730 

1036 

1259 

1429 

15    0 

1510 

356 

735 

1043 

13    8 

1439 

1510 

1520 

358 

740 

1051 

1317 

1449 

1520 

1530 

4    ] 

745 

1058 

1325 

1458 

1530 

1540 

4    3 

750 

11    5 

1334 

15    8 

1540 

1550 

4    6 

755 

1112 

1343 

1518 

1550 

16    0 

4    8 

8    0 

1119 

1351 

1527 

16    0 

1610 

411 

8    5 

1126 

14    0 

1537 

1610 

1620 

414 

810 

1133 

14    9 

1547 

1620 

1630 

416 

815 

1140 

1417 

1556 

1630 

1640 

419 

820 

1147 

1426 

16    6 

1640 

1650 

421 

825 

1154 

1435 

1616 

1650 

17    0 

424 

830 

12    1 

1443 

1625 

17    0 

1710 

427 

835 

12    8 

1452 

1635 

1710 

1720 

429 

840 

1215 

15    1 

1645 

1720 

1730 

432 

845 

1222 

15    9 

1654 

1730 

1740 

434 

850 

1230 

1518 

17    4 

1740 

1750 

437 

855 

1237 

1527 

1714 

1750 

18    0 

440 

9    0 

1244 

1535 

1723 

18    0 

1810 

442 

9    5 

1251 

1544 

1733 

1810 

1820 

445 

910 

1258 

1553 

1743 

1820 

1830 

447 

915 

13    5 

16    1 

1752 

1830 

1840 

450 

920 

1312 

1610 

18    2 

1840 

1850 

452 

925 

1319 

1619 

1812 

1850 

19    0 

455 

930 

1326 

1627 

1821 

19    0 

1910 

458 

935 

1333 

1636 

1831 

1910 

1920 

5    0 

940 

1340 

1645 

1840 

1920 

1930 

5    3 

945 

1347 

1653 

1850 

1930 

1940 

5    5 

950 

1354 

17    2 

19    0 

1940 

1950 

5    8 

9-55 

14    1 

1711 

19    9 

1950 

724  NAUTICAL  ASTRONOMY 

TABLE  IV. — Products  of  Arcs 
Multiplied  by  the  Sines  of  15°  Rhumbs. —  (continued). 


ARCS. 

fifi 

Sin.  15° 

S2 

Sin.  30° 

S3 

Sin.  45' 

S4 

Sin.  60° 

s,  . 

Sin.  75° 

ARCS. 

0         , 

0            1 

0            / 

0           f 

0             , 

0           / 

0         / 

20    0 

511 

10    0 

14    9 

1719 

1919 

20    0 

2010 

513 

10    5 

1416 

1728 

1929 

2010 

2020 

516 

1010 

1423 

1737 

1938 

2020 

2030 

518 

1015 

1430 

1745 

1948 

2030 

2040 

521 

1020 

1437 

1754 

1958 

2040 

2050 

524 

1025 

1444 

18    3 

20    7 

2050 

21    0 

526 

1030 

1451 

1811 

2017 

21    0 

21  10 

529 

1035 

1458 

1820 

2027 

21  10 

21  20 

531 

1040 

15    5 

1829 

2036 

21  20 

21  30 

534 

1045 

1512 

1837 

2046 

21  30 

21  40 

536 

1050 

1519 

1846 

2056 

2140 

21  50 

539 

1055 

1526 

18  54 

21    5 

21  50 

22    0 

542 

11    0 

1533 

19    3 

2115 

22    0 

2210 

544 

11    5 

1540 

1912 

2125 

2210 

2220 

547 

1110 

1548 

1920 

2134 

2220 

2230 

549 

1115 

1555 

1929 

2144 

2230 

2240 

552 

1120 

16    2 

1938 

2154 

2240 

2250 

555 

1125 

16    9 

1946 

22    3 

2250 

23    0 

557 

1130 

1616 

1955 

2213 

23    0 

2310 

6    0 

1135 

1623 

20    4 

2223 

2310 

2320 

6    2 

1140 

1630 

2012 

2232 

2320 

2330 

6    5 

1145 

1637 

2021 

2242 

2330 

2340 

6    8 

1150 

1644 

2030 

2252 

2340 

2350 

610 

1155 

1651 

2038 

23    1 

2350 

24    0 

613 

12    0 

1658 

2047 

2311 

24    0 

2410 

615 

12    5 

17    5 

2056 

2321 

2410 

2420 

618 

1210 

1712 

21    4 

2330 

2420 

2430 

620 

1215 

1719 

2113 

2340 

2430 

2440 

623 

1220 

1727 

2122 

2350 

2440 

2450 

626 

1225 

1734 

2130 

2359 

2450 

PRODUCTS  OF  ARCS  BY  SINES 

TABLE  IV. — Products  of  Arcs 
Multiplied  by  the  Sines  of  15°  Rhumbs. —  (continued). 


725 


AECS. 

B, 

Sin.  15° 

S2 

Sin.  30° 

S3 

Sin.  45° 

S4   - 

Sin.  60° 

S5 

Sin.  75° 

AECS. 

0        / 

0           / 

0           1 

0           t 

0           / 

O           1 

-     / 

25    0 

628 

1230 

1741 

2139 

24    9 

25    0 

2510 

631 

1235 

1748 

2148 

2418 

2510 

2520 

633 

1240 

1755 

2156 

2428 

2520 

2530 

636 

1245 

18    2 

22    5 

2438 

2530 

2540 

639 

1250   • 

18    9 

2214 

2448 

2540 

2550 

641 

1255 

1816 

2222 

2457 

2550 

26    0 

644 

13    0 

1823 

2231 

25    7 

26    0 

2610 

646 

13    5 

1830 

2240 

2517 

2610 

2620 

649 

1310 

1837 

2248 

2526 

2620 

2630 

652 

1315 

1844 

2257 

2536 

2630 

2640 

654 

1320 

1851 

23    6 

2545 

2640 

2650 

657 

1325 

1858 

2314 

2555 

2650 

27    0 

659 

1330 

19    6 

2323 

26    5 

27    0 

2710 

7    2 

1335 

1913 

2332 

2614 

2710 

2720 

7    4 

1340 

1920 

2340 

2624 

2720 

2730 

7    7 

1345 

1927 

2349 

2634 

2730 

2740 

710 

1350 

1934 

2358 

2643 

2740 

2750 

712 

1355 

1941 

24    6 

2653 

2750 

28    0 

715 

14    0 

1948 

2415 

27    3 

28    0 

2810 

717 

14    5 

1955 

2424 

2712 

2810 

2820 

720 

1410 

20    2 

2432 

2722 

2820 

2830 

723 

1415 

20    9 

2441 

2732 

2830 

2840 

725 

1420 

2016 

2450 

2741 

2840 

2850 

728 

1425 

2023 

2458 

2751 

2850 

29    0 

730 

1430 

2030 

25    7 

28    1 

29    0 

2910 

733 

1435 

2037 

2516 

2810 

2910 

2920 

736 

1440 

2045 

2524 

2820 

2920 

2930 

738 

1445 

2052 

2533 

2830 

2930 

2940 

741 

1450 

2059 

2542 

2839 

2940 

2950 

743 

1455 

21    6 

2550 

2849 

2950 

726  NAUTICAL  ASTRONOMY 

TABLE  IV. — Products  of  Arcs 
Multiplied  by  the  Sines  of  15°  Rhumbs. —  (continued). 


ARCS. 

Si 

Sin.  15° 

S2 

Sin.  30° 

Sa 

Sin.  45° 

S4 

Sin.  60' 

S5 

Sin.  75" 

ARCS. 

0         / 

0           1 

0            / 

0           / 

0          / 

0            / 

0        , 

30    0 

746 

15    0 

2113 

2559 

2859 

30    0 

3010 

748 

15    5 

2120 

26    8 

29    8 

3010 

3020 

751 

1510 

2127 

2616 

2918 

3020 

3030 

754 

1515 

2134 

2625 

2928 

3030 

3040 

756 

1520 

2141 

2633 

2937 

3040 

3050 

759 

1525 

2148 

2642 

2947 

3050 

31    0 

8    1 

1530 

2155 

2651 

2957 

31    0 

31  10 

8    4 

1535 

22    2 

2659 

30    6 

31  10 

3120 

8    7 

1540 

22    9 

27    8 

3016 

31  20 

31  30 

8    9 

1545 

2216 

2717 

3026 

31  30 

3140 

812 

1550 

2224 

2725 

3035 

31  40 

31  50 

814 

1555 

2231 

2734 

3045 

31  50 

32    0 

817 

16    0 

2238 

2743 

3055 

32    0 

3210 

820 

16    5 

2245 

2751 

31    4 

3210 

3220 

822 

1610 

2252 

28    0 

3114 

3220 

3230 

825 

1615 

2259 

28    9 

3124 

3230 

3240 

827 

1620 

23    6 

2817 

3133 

3240 

3250 

830 

1625 

2313 

2826 

3143 

3250 

33    0 

832 

1630 

2320 

2835 

3153 

33    0 

3310 

835 

1635 

2327 

2843 

32    2 

3310 

3320 

838 

1640 

2334 

2852 

3212 

3320 

3330 

840 

1645 

2341 

29    1 

3222 

3330 

3340 

843 

1650 

2348 

29    9 

3231 

3340 

3350 

845 

1655 

2355 

2918 

3241 

3350 

34    0 

848 

17    0 

24    3 

2927 

3250 

34    0 

3410 

851 

17    5 

2410 

2935 

33    0 

3410 

3420 

853 

1710 

2417 

2944 

3310 

3420 

3430 

856 

1715 

2424 

2953 

3319 

3430 

3440 

858 

1720 

2431 

30    1 

3329 

3440 

3450 

9    1 

1725 

2438 

3010 

3339 

3450 

PRODUCTS  OF  ARCS  BY  SINES 

TABLE  IV. — Products  of  Arcs 
Multiplied  by  the  Sines  of  15°  Rhumbs. —  (continued). 


727 


ARCS. 

St 

Sin.  15° 

S2 

Sin.  30° 

S3 
Sin.  45° 

S4 

Sin.  60° 

S5 

Sin.  75° 

ARCS. 

0        / 

0           / 

o        r 

0           / 

0           / 

0           1 

O        1 

35    0 

9    4 

1730 

2445 

3019 

3348 

35    0 

3510 

9    6 

1735 

24  52 

3027 

3358 

3510 

3520 

9    9 

1740 

2459 

3036 

34    8 

3520 

3530 

911 

1745 

25    6 

3045 

3417 

3530 

3540 

914 

1750 

2513 

3053 

3427 

3540 

3550 

916 

1755 

2520 

31    2 

3437 

3550 

36    0 

919 

18    0 

2527 

3111 

3446 

36    0 

3610 

922 

18    5 

2534 

3119 

3456 

3610 

3620 

924 

1810 

2541 

3128 

35    6 

3620 

3630 

927 

1815 

2549 

3137 

3515 

3630 

3640 

929 

1820 

2556 

3145 

3525 

3640 

3650 

932 

1825 

26    3 

3154 

3535 

3650 

37    0 

935 

1830 

2610 

32    3 

3544 

37    0 

3710 

937 

1835 

2617 

3211 

3554 

3710 

3720 

940 

1840 

2624 

3220 

36    4 

3720 

3730 

942 

1845 

2631 

3229 

3613 

3730 

3740 

945 

1850 

2638 

3237 

3623 

3740 

3750 

948 

1855 

2645 

3246 

3633 

3750 

38    0 

950 

19    0 

2652 

3255 

3642 

38    0 

38  10 

953 

19    5 

2659 

33    3 

3652 

3810 

3820 

955 

1910 

27    6 

3312 

37    2 

3820 

3830 

958 

1915 

2713 

3321 

3711 

3830 

3840 

10    0 

1920 

2720 

3329 

3721 

3840 

3850 

10    3 

1925 

2728 

3338 

3731 

3850 

J28 


NAUTICAL  ASTRONOMY 


TABLE  V. 

DISTANCES  PROM  COMPASS  AT  WHICH  QTTADRANTAL  CORRECTORS 
SHOULD  BE  PLACED  FOR  VALUE  OF  D. 


FOB  BINNACLES  OF  TYPE  VI. 


Distance  from 

V  il  1  III/  UJ.    U. 

Compass 
on  Graduated 
Quadrantal  Arms. 

For  7-Inch 
Spheres. 

For  9-Inch 
Spheres. 

For  Filled 
Chain  Boxes. 

Inches. 

•o         / 

0             t 

o          t 

11 

12     00 

21     15 

11.5 

9     15 

19     00 

10     30 

12 

7     45 

17     00 

8     45 

12.5 

6     45 

14     15 

7     30 

13 

5     45 

12     00 

6     15 

13.5 

4     45 

10     00 

5     15 

14 

4     00 

8     45 

4     20 

14.5 

3     30 

7     30 

4     00 

15 

3     00 

To  find  D  when  $>  is  given,  use  formula  :  D  =  2)  X  67°.3. 


EXTRACTS  FROM  N.  A    1905 


729 


l 

P'-'E 


H  ^ 


<M*  O  00  CO  O  W  00  CJ  t-* 


mi- 
ete 


b-  O  O  O  kO 


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eq(»  jo    * 


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COTHOOrJH 
"*  O          rH  rH  (M  75 


O  O  O 
>O          »H 


•q^  jo 


?30 


NAUTICAL  ASTRONOMY 


«C  OS  «O  (M  <M 
^  (M  CXI  Cd  i-H 

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5  rH  £ 
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apparent  n 
t  south  decl 


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eclination  indicates 


ter  for  mean  noon 
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EXTRACTS  FROM  N".  A,  1905 


731 


732 


NAUTICAL  ASTRONOMY 


cl<OO-»*cOO?«O©COO?<Ot-TH 
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EXTRACTS  FROM  N.  A.  1905 


733 


734 


NAUTICAL  ASX-RONOMY 


H 

II 


Meridian  o 
Greenwich 


O  M 

s'l 


S'w 

fir-, 


TH  ^-  <N  c5  <M  o  <M  CM  s?  w 


co  r>  o  o  r-  t    --i  j>  co  as  i> 


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EXTRACTS  FROM  N.  A.  1905 


"35 


CO    O?    5D   r4  i— ('  i-H    T— I    O 


CO          CO  OS  •*  O>  t-  •*  CO  t- 


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Ascension. 


oj  10  ,-4  r-  us 


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4- 


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i-diameter 
Horizontal  Parallax 


Semi 


736 


NAUTICAL  ASTRONOMY 


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EXTRACTS  FROM  .N.  A.  1905 


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NAUTICAL  ASTRONOMY 


o  ;±<:3ic5'-MHiH;- 

++ 1  + 1  ++ 1 '+ 1  I  I +++ 


i-HOOi-Hi-HOoioeii-ii-ii-ii-HOrH 

I    I 


ja  a 


s-s 


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ft 


U.  S.  Hydrographic  Office  Charts— Conventional  Signs  and  Symbols 


PLATE  X. 


LigTtt  House,  or  Lighted,  Beacon- 
Light  Vessels..*.  ______  ...........  ,  ..................... 

BeUBoat.-  ..........  «  .........................  .  ................  A- 

Beacons  /not  UghtedJ.:,....-.&r,*.L  i  i  I  I  I 

Spindle  or  stake-.  ..........  ----  ...........................  i 

flooring  "buoy......  .................  ......  .,  ........  ^.  .......  ™t£» 

Green,  red.yeUow,  or  white  buoy.  ..............  ; 

flack  buoy.  .....................................  :....,  ..........  »..t 

Danger  buoy  (horizontal,  stripes)  ---------  .,J? 

Channel  b'uoy  (vertical  stripes)  ,  ................  f 

Whistling  buoys  ..........................................  §  f  f  f 

Bell  buoys,...  .......................  ,  .........................  \  \  \  \ 

Lighted,  buoys  ...........  ....  .......  „ 

Distinctive  "buoys 
Wreck  .............  ........  , 

Life,  saving  station: 


§  f  0  f 
\\\\ 


Anchorage  for  large  vessels  ~  .....  _.r  ........  .  .«!> 

Anchorage  for  small  vessels.  ......................  .f> 

Rock  above  -water  ........  ,  ...........  .  ............  .,.-..,€!  »  ® 

Rock  underwater  ........  „  ........  _  ...................  ,»    £« 

Rock  atvash  at  any  stage  of  the  tide.*    (*} 
Rock  -whose  position  is  doubtful  ._^...{£  P.D 
Rock  "whose*  existence  is  doubtful  .........  ®  £D 

Jfo  bottom,  at  50  fathoms 
'Currents,  velocity  .2  knots  ..............  ,..._; 

(Flood  it  Taiots,. 

\Ebbl  knot,  ....... 

2d.  hour  flood  current  ol.  ,., 

3d,,  hour  ebb  current  .......        ...  »  •"•         ,,,  * 

The  period  of  a  tidal  current  and  its  direc- 
tion is  sometimes  denoted  by  I  Qr.,  If  Qr., 
etc*,  or  lh.,11  h..etc.,  on  the,  arrow  thus: 
3d*  Quarter,  flood  current.  .................  _»'  Q'  , 

1st.  quarter  ebb  current  ......................      '  Qr  , 

2d.  hour  flood  current  ......................  ,  _  11  n    t 

4th  hour  ebb  current  w  r> 


Tidal  Currents 


Cities  (according  to  scale  of  chart; 

Towns  and  "Tillages  (according  to  scale,)  J^  -tt  o 

Single-  houses — t. - *  •  sa 

Churches ~ -~ +  E  4 

Fort  or  Battery. +   f* 

Windmill.- ... _ _..j5  8 

Observation  spot,. „ e 

Single  trees  and  groups... i 

Cemetery.. ; - ^  •"•  or 


Fences  and  Sedges  ...  .........  -- 

Triangulation  station 
Flagstaff.  ..........  „ 

Semaphore-  or  Signal  station 
Storm,  signal  station 


Dam,  ......  „ 


Fish  weirs. 


In  locating  the,  symbol  for  lighthouses 
and  lightvessels  on  charts  the  Ught  dot 
is  placed  in  the-  geographical  position 
assigned  to  the  lighthouse  or  Ughtvessel. 
When,  on  a  Ughtvessel  there  are 
more  lights  than  one.  either  on  the.  same 
or  on  different  masts,  the  middle  of  the 
line  joining  the  centers  of  the  first  and 
last  dots  is  placed  in  the  geographical 
position  assigned  to  the  lightvcssel. 

When  there  is  no  explanatory  note  refer- 
ring to  the  buoys  on  a  chart  their  color  is 
indicated  by  "words  or  their  abbrevuitions 
placed  near  the  buoy  The  ring  at  .the 
end  of  all  buoys,  and  the  middle  of  tfiA 
base  line  of  the  symbol  for  beacons  is 
placed  in  the  geographical  position  assignr 
ed.  to  the,  buoy  or  beacon. 


U.  S.  Coast  and  Geodetic  Survey  Charts— Hydrographic  Signs  PLATE  XI. 


Lighthouse r^ 

Mighthou»e  on.  small  scale  chart  •       OlcLlight  tourer ® 

^Beacon,  lighted, &     IBeajcan,-not,'lighte.d A 

Spindle  (or  atcJte  ) \       add  word.  Spindle  if  frpace  allows . 

JLightship &     Wreck -*• 

Anchorage ."*»      Covering  and.  -uncovering  rock. 0 

JRocTt  a-wash,  at  low    wcvter *      Sunken,  roclz -v 

Salving  Station +  L.S.S.    (T) signifies  connection,  yri£i. 

•telegraphic  system. 


27b  bottom  at,  2O  fathoms , 

buoy     V      or  add,  word,    -white    or  yellow  as 

buoy...~ • 

^Horizontally   striped,  buoy • 

Perpendicularly   striped,  buoy * 

iiH 

J3uf>ys    with    perch,    and,    $cju.c.re 

jBuoys    -witsL    perch,    and,    bcJi 

Xighted  buoy... ^  ,    in  plaee  of    o  _,    a.s \ 

2xu)orin£i  buoy. < 

Landmark .  o^s  Cupola,,  Standpipe ,  etc 

Whirlpool 

Ode  rip 1. 

Current,  not  tidal ,  drift  in  Ttnots  a.3 »— — —  2.O 

"    fflood,,Ttrat  au&rter .  drift  in,  Jcn&ts,  CLS — 0.^ 

l.O 

0.3 


bb  ...  , ^  other-wise   bO^e.    flood.. 


U    S.  Coast  and  Geodetic  Survey  Charts— Topographic  Signs, 


PLATE  XII. 


'Xtyw  Water 


Rocky  Ledges 


Roclty  Bluff 


Eroded  JBanh 


Sand  and  STungl* 


Palms 


OaJ* 


JDecidiu>ua  and 
Undergrowth 


Pint 


Cactt 


U.  S.    Coast  and  Geodetic  Survey  Charts— Topographic  Signs. 


PLATE  XIII 


Cypress  Swamp 


Groat 


RiceDikea  AIHtchcs 


Wooded  JtfarsK 


Submerged  Marsh    =•  ^J^  =? 


Curves  of  equal 

elevation  and 

intermediate  curves 


JScl  Grass 
EXp 


PLATE  XIV. 


GENERAL  ABBREVIATIONS 

ON  HYDROGRAPHIC  OFFICE  CHARTS  AND  U.  S.  COAST  AND 
GEODETIC  SURVEY  CHARTS. 


ABBREVIATIONS  FOR  KINDS  OF  BOTTOM. 


M Mud. 

S Sand. 

G Gravel. 

Sh Shells. 

P Pebbles. 

Sp Specks. 

Cl Clay. 

St Stones. 

Co Coral. 

Oz..          ..Ooze. 


bk Black. 

wh White. 

rd Red. 

yl Yellow. 

gy Gray. 

bu Blue. 

dk Dark. 

It Light. 

gn Green. 

br Brown. 

ABBREVIATIONS  NEAR  BUOYS. 


hrd Hard. 

sft Soft. 

fne Fine. 

crs Coarse. 

brk Broken. 

Irg Large. 

sml Small. 

rky Rocky. 

stk Sticky. 

stf..          ..Stiff. 


U.  S.  C.  and  G.  S.  Charts. 

C Can. 

N Nun. 

S Spar. 


Hydrographic  Office  Charts. 

B,  bk.... Black.         Y,  yl Yellow, 

W,wh White.        Ch,  chec Checkered. 

R,  rd Red.  H.S . . Horizontal  stripes. 

G,  gn Green.         V.S Vertical  stripes. 


F 

Fig... 

Fl 

Fls... 


Fixed. 
Flashing. 
Flash. 
Flashes. 


ABBREVIATIONS  FOR  LIGHTS. 

Rev Revolving.  V. 

E Eclipses. 

W White. 

R. .        . .  Red. 


. . .  Varied  by. 
Sec Sector. 


Bn. . .  .Beacon.         L.  W Low  water. 

kn Knots.  H.  W High  water. 

H.  W.  F.  &  C High  water  at  full  and 

change  of  moon. 


L.  S.  S . .  Life-saving  station. 

P.  D Position  doubtful. 

E.  D  . .  .Existence  doubtful. 


A  wireless  station  is  indicated  by  a  point  in  a  circle  and  the  legend 
"Wireless  Station."  The  information  as  to  a  submarine  bell  is  covered 
by  the  legend  "Submarine  Bell"  at  the  spot,  and,  in  case  of  a  lightship, 
in  the  table. 


PLATE  XV. 


(1)  Pole  without  the  circle,  curve  is  closed;  as  CC'C",  DD'D". 

(2)  Pole  within  the  circle,  curve  is  sinusoidal ;  as  AA'A". 

(3)  Pole   on   the   circle,    curve   is  open,   branches   meeting  at   infinity,   asymptotos 

parallel  to  meridians ;  as  BB'B". 


APPENDIX    A. 

DESCRIPTION  OF  THE  SUBMARINE-BELL  SYSTEM. 

The  equipment  furnished  by  the  submarine-bell  company  to  a 
vessel  or  station  depends  on  the  purposes  for  which  intended,  and 
may  therefore  be  considered  under  the  two  general  heads  given 
below. 

The  sending  apparatus,  consisting  of  the  bell  and  accessories, 
varies  according  to  the  use  made  of  the  system.  When  installed 
on  light  ships  and  tenders,  the  outfit  consists  of  a  bell  mounted 
on  a  case  containing  the  striking  mechanism  which  is  operated  by 
compressed  air  supplied,  through  a  hose,  from  air  tanks;  a  davit, 
with  chain  and  windlass,  for  raising  or  lowering  the  bell  over  the 
ship's  side;  and  a  code  ringer  for  so  controlling  the  strokes  as  to 
make  automatically  the  code  number  of  the  light  ship. 

For  use  near  certain  dangers  or  turning  points,  the  bell  is  hung 
on  a  tripod  standing  on  the  bottom,  and  the  clapper  is  actuated 
by  powerful  magnets  energized  by  a  current  sent  from  a  shore 
power-house,  through  an  armor-protected  submarine  cable. 

When  suspended  from  buoys  the  bell  has  a  mechanism  consist- 
ing of  a  combination  of  ratchets  and  pawls  through  whose  agency 
a  spring  is  compressed  to  a  certain  point  by  the  wave-action  on 
the  buoy,  and  then  automatically  released,  causing  the  clapper  to 
strike  the  bell.  \ 

The  receiving  apparatus  installed  in  vessels  consists  of  two  small 
tanks  placed  in  the  forward  part  of  the  vessel,  well  below  the 
water-line,  one  against  the  starboard  side,  one  against  the  port 
side.  In  each  tank  are  two  microphones  immersed  in  liquid 
which  receive  the  sounds,  when  the  sound-waves  strike  the  ship's 
side.  These  sounds  are  transmitted  to  the  pilot-house,  or  other 
location  of  the  direction-indicator.  Wires  are  run  from  the  tanks 
to  the  battery  box  which  supplies  the  power,  thence  to  the  direc- 
tion-indicator which  is  a  small  round  metallic  case  fastened  to 
the  wall  with  telephone  receivers  hung  on  each  side,  and  bearing 
on  its  face  a  switch  for  connecting  either  starboard  or  port 
microphones  with  the  receivers;  a  dial  indicates  the  one  connected. 


T-iS 


APPENDIX    B. 


COMPENSATION   OF  COMPASSES   AT  A   SHIP-YARD  BEFORE 
PROCEEDING  TO   SEA.* 

Before  a  vessel  is  sent  to  sea  for  the  first  time  from  a  dock-yard 
or  a  navy-yard,  the  navigator  should,  by  a  preliminary  compen- 
sation, so  reduce  the  deviations  and  equalize  the'  directive  force 
of  the  compasses  that  they  may  be  used  to  steer  by  until  he  shall 
be  able  to  compensate  them  regularly  and  obtain  a  residual  curve. 

This  preliminary  compensation  should  always  be  made,  when 
possible,  from  data  obtaine'd  from  observations  and  vibrations  on 
two  headings,  assuming  51  and  (£  as  zero,  and  determining  j$,  (f, 
and  ®  by  the  method  of  Art.  95  if  computation  is  made,  as  it 
should  be  whenever  construction  would  give  acute  angles  of  inter- 
section, or  by  the  method  of  Art.  113  if  the  dygogram  is  used. 

Then  compensation  -should  be  made  as  explained  in  Art.  110; 
or,  by  using  the  indications  of  the  dygogram,  neutralizing  first 
the  quadrantal  force  and  then  in  the  proper  order  as  shown  by 
the  dygogram  the  semicircular  forces.  As  each  corrector  is 
placed,  the  deviation  should  be  reduced  to  the  amount  indicated 
by  a  dygogram  of  the  remaining  force  or  forces  only. 

Provided  the  compasses  are  uninfluenced  by  the  presence  of 
other  vessels,  structures  or  masses  of  steel  or  iron,  the  required 
observations  may  be  obtained: 

(1)  When  the  vessel  is  in  drydock  and  also,  at  an  earlier  or 
later  time,  alongside  a  dock  or  sea-wall. 

(2)  When  moored  alongside  a  dock  or  sea-wall  and  the  ship 
can  be  either  winded  or  sprung  out  to  a  suitable  heading   (see 
Arts.  95,  110,  and  113). 

Whilst  X,  23,  d,  and  2)  may  all  be  determined  when  observa- 
tions on  two  headings  are  possible,  it  may  sometimes  happen  that 
these  can  be  made  on  only  one  heading.  Under  such  circum- 
stances X  and  ®  must  be  assumed,  and,  for  reasons  given  in 
Art.  95,  they  may  be  assumed  as  those  of  a  similarly  situated 
compass  on  a  similar  ship. 

Make  the  necessary  observations  and  vibrations  referred  to  in 
Art.  94  before  the  quadrantal  spheres  are  placed  and  determine 
23  and  &  by  computation,  using  equations  (69a)  and  (70a);  or  by 

*  Ser-  "  Tlio  First  Compensation  of  a  Vessel's  Compasses,"  issued  by  Bureau  of 
Equipment,  Navy  Department,  IUOG. 


APPENDICES  749 

construction  as  explained  in  Art.  113a.  Then  having  23,  (£,  and  ©, 
proceed  to  compensate  as  directed  in  Art.  110.  If  these  values  are 
found  by  construction,  Art.  113a,  the  dygogram  may  be  used 
when  compensating  to  indicate  the  deviation  and  the  changes  in 
deviation  as  each  force  is  successively  neutralized. 

SPECIAL  PROCEDURE  WHEN  THE  SHIP  IS  ON  CERTAIN 
HEADINGS. 

Special  procedure  for  compensation  of  the  compass,  ship  head- 
ing on  a  cardinal  point  magnetic  (assuming  21  and  &  as  zero). 

Having  obtained  the  coefficients  23,  (£,  and  S>  by  any  of  the 
methods  explained  in  Chapter  IV,  it  may  happen,  when  compen- 
sation takes  place  on  one  heading,  as  contemplated  in  Art.  94, 
that  the  vessel  is  unavoidably  heading  with  the  keel-line  north 
and  south  magnetic  or  east  and  west  magnetic.  Again,  the  vessel 
may  be  on  the  stocks  or  alongside  a  dock  with  the  keel-line  as 
indicated  above  and  compass  coefficients  unknown,  when  it  be- 
comes necessary  to  assume  X  and  &  as  those  of  a  compass  simi- 
larly situated  on  a  sister  ship,  to  determine  93  and  (£,  and  then  to 
compensate  the  compass  (see  Arts.  94  and  113a).  In  such  cases 
compensation  for  two  of  the  forces  according  to  Art.  110  may 
appear  indeterminate  and  special  procedure  becomes  necessary. 

(1)  Let  the  heading  be  assumed  as  magnetic  north. — On  this  head- 
ing neither  93  nor  <D  produces  any  deviation  but  they  both  influence 
the  amount  produced  by  (£.  With  the  given  or  assumed  ©,  find 
D  =  3)  X  57°. 3;  for  this  value  of  D  take  from  Table  V  the  dis- 
tance at  which  the  quadrantal  spheres  should  be  placed  to 
neutralize  the  quadrantal  force  and  so  place  them  on  the  arms. 
This  eliminates  ©.  With  <D  neutralized,  it  is  evident  that  the 
deviation  produced  by  (£  is  still  influenced  by  the  force  93  acting 
in  the  fore  and  aft  line.  Were  it  not  for  the  influence  of  23,  Cf 
would  produce  a  deviation  equal  to  tan-i  (£,  and  if  the  forces  act- 
ing in  the  fore  and  aft  line  should  be  so  altered  that  a  deviation 
of  tan-i  (£  should  be  shown  by  the  compass,  the  force  23  would 
then  be  compensated. 

Therefore,  run  the  fore  and  aft  carrier  down,  fill  tubes  with 
magnets,  red  ends  forward  if  23  is  +,  aft  if  23  is  ( — ) ;  raise 
carrier  slowly  till  the  required  deviation,  tan-i  (£  (angle  POC" 
in  Figs.  149  and  150),  is  indicated,  and  clamp  the  carrier;  the 
force  23  is  thus  neutralized. 


750 


APPENDICES 


Run  the  athwartship  carrier  down;  fill  tubes  with  magnets  and 
place  them  red  ends  to  starboard  if  &  is  +,  to  port  if  (£  is  ( — ) ; 
raise  carrier  slowly  till  compass  points  north  magnetic,  which  is 
the  ship's  heading;  the  force  (£  is  thus  neutralized  and  the 
compass  is  compensated. 

The  steps  above  taken  may  be  illustrated  by  the  dygogram 
(Figs.  149  and  150).  Let  OP  —  unity,  PD'  =  <&,  D'B  =  93,  BC  =  $; 


both  93  and  (£  being  -+-  in  Fig.  149  and  —  in  Fig.  150.  Though 
93  and  ©  produce  no  deviation  on  this  heading,  magnetic  north, 
they  influence  the  amount  produced  by  &;  the  deviation  under 
the  various  influences  being  POC  and  the  direction  of  the 
needle  00. 

When  S>  is  compensated,  PD'  shortens  to  zero,  D'B  and  BG 
respectively  assume  the  positions  PB'  and  B'C'  and  the  needle 
takes  the  position  00',  a  position  it  should  assume  when  under 
the  influences  only  of  93  and  (£.  If  93  were  then  compensated,  the 
line  PB'  would  shorten  to  zero  and  B'C'  would  assume  the  posl- 


APPENDICES 


751 


tion  PC",  the  needle  would  take  the  direction  00",  and  the 
deviation  would  become  POC"  (which  angle  equals  tan-i  (£) 
and  be  due  to  (£  alone  uninfluenced  by  any  other  of  the  ship's 
forces;  therefore,  after  <£>  has  been  eliminated  by  placing  the 
spheres  according  to  Table  V,  and  only  the  semicircular  forces 
remain  acting,  ship  heading  north  magnetic,  place  the  fore-and- 
aft  corrector-magnets  so  as  to  alter  the  deviation  from  POC'  to 


POC",  then  place  the  athwartship  corrector-magnets  to  reduce  the 
deviation  from  POC"  to  zero;  the  needle  will  take  the  direction 
of  OP  and  the  compass  will  be  compensated. 

(2)  Let  the  heading  be  assumed  as  magnetic  east.— In  this  case, 
23  produces  the  deviation,  the  amount  of  which,  however,  is  in- 
fluenced by  the  forces  (£  and  2)  acting  athwartship  and  in  the 
magnetic  meridian;  therefore,  the  procedure  should  be  as  follows: 
Place  quadrantal  spheres  as  indicated  above,  neutralizing  $>; 
by  means  of  the  athwartship  magnets  so  alter  the  athwartship 
forces  that  the  compass  will  indicate  a  deviation  of  tan-i  93  (angle 


752 


APPENDICES 


FOB',  Figs.  151  and  152),  thus  neutralizing  (£;  then,  by  means  of 
magnets  properly  placed  in  the  fore-and-aft  carrier,  make  the 
compass  indicate  the  heading  east  magnetic,  eliminating  the  force 
83  and  completing  the  compensation  of  the  compass. 

The  steps  above  taken  when  the  ship  headed  east  magnetic 
may  be  illustrated  by  the  dygograms  (Figs.  151  and  152);  both 
ft  and  (£  being  +  in  Fig.  151  and  —  in  Fig.  152. 

When  S)  has  been  neutralized  PD  shortens  to  zero,  DB  and  BC 


respectively  assume  the  positions  PB'  and  B'C',  the  deviation  be- 
comes POC',  and  the  needle  takes  the  direction  00'.  When  &  has 
been  neutralized,  B'C'  shortens  to  zero,  the  needle  lies  in  direction 
OB',  the  deviation  becomes  POB'  (which  angle  equals  tan-i  93) 
and  is  due  to  23  alone  uninfluenced  by  any  other  of  the  ship's 
forces.  If  this  value  is  reduced  to  zero  by  fore-and-aft  corrector- 
magnets  properly  placed,  the  compass  needle  should  take  the 
direction  OP  and  the  compensation  be  effected. 

(3)  With  S>  eliminated  and  ship  heading  east  or  west  per  com- 
pass.— if  the  heading  is  such  that  after  the  quadrantal  force  has 


APPENDICES 


753 


been  eliminated,  the  ship  should  be  heading  east  per  compass 
when  the  deviation  is  easterly,  or  west  per  compass  when  the 
deviation  is  westerly,  the  athwartship  force  (£  will  lie  in  the 
vertical  plane  through  the  compass  needle,  and,  as  shown  by  the 
dygograms  (Figs.  153  and  154),  the  corresponding  corrector- 
magnets,  if  used  first,  would  have  no  apparent  effect  when  com- 
pensating. However,  knowing  the  deviation  and  compass  head- 
ing, we  may  easily  find  the  magnetic  heading  and  the  amounts  of 


deviation  produced  respectively  by  the  forces  23  and  (£  on  that 
heading;  after  which,  the  compensation  by  the  method  of  Art.  110 
is  very  simple,  for  the  force  'ft,  being  more  nearly  at  right  angles 
with  the  needle,  should  be  eliminated  first  and  then  the  force  (£. 
If  not  wishing  to  compute  the  deviation  due  to  these  forces, 
nor  to  apply  the  method  of  Art.  110,  we  may  construct  a  dygo- 
gram  and  use  it  in  compensating.  The  dygograms  (Pigs.  153 
and  154)  show  that  if  the  force  93  should  be  compensated,  PB 
would  shorten  to  zero,  the  needle  would  assume  the  position  OC", 
the  deviation  would  become  POC',  and  the  remaining  force  (£ 


754 


APPENDICES 


would   be   left   at   a   favorable   angle   with   the   direction   of   the 
needle    for    its    elimination.     Therefore,    place    the    fore-and-aft 


153. 


FIG. 


0 

154. 


corrector-magnets  at  such  a  height  as  to  change  the  deviation 
from  FOB  to  POC',  compensating  the  force  £5;    then  place  the 


APPENDICES 


755 


athwartship  corrector-magnets  at  such  a  height  as  to  reduce  the 
deviation  from  POC'  to  zero,  compensating  the  force  (£,  and 
thereby  effecting  the  complete  compensation  of  the  compass. 

(4)  On  a  heading  of  no  semicircular  deviation. — If  the  heading  of 
the  ship  is  such  that,  after  the  quadrantal  force  has  been  elimi- 
nated, no  deviation  is  shown  when  S3  and  (£  are  known  to  have 
appreciable  values,  then  it  is  evident  that  the  semicircular  forces 
are  neutralizing  each  other  on  that  particular  heading.  This 


O 

FIG.  155. 

state  of  affairs  is  indicated  by  the  dygogram  (Fig.  155),  the 
compass  needle  there  lying  in  the  meridian  OP. 

The  method  employed  in  Art.  110  for  the  elimination  of  S3  and 
(£  may  be  followed  in  this  case  and  the  compensation  of  the 
compass  effected  without  any  difficulty. 

However,  if  not  wishing  to  compute  the  deviations  due  to  S3 
and  (£,  the  dygogram  may  be  used  in  compensating  as  indicated 
below;  at  all  events,  it  will  serve  the  useful  purpose,  as  it  does  in 
all  cases  when  used,  of  indicating  which  force  it  is  preferable  to 
eliminate  first. 

For  the  particular  values  of  S3  and  (£  indicated  by  Fig.  155,  it 


756  APPENDICES 

is  evident  that  it  would  be  better  to  eliminate  first  the  force  (£, 
shortening  BC  to  zero,  causing  the  needle  to  lie  in  the  direction 
OB,  and  leaving  the  remaining  force  23  at  an  angle  with  the 
direction  of  the  needle  more  favorable  for  its  elimination  than 
would  have  been  the  angle  for  the  elimination  of  (£  had  the  force 
93  been  neutralized  first. 

Therefore,  place  the  athwartship  magnet-correctors  at  such  a 
height  as  to  produce  a  deviation  equal  to  the  angle  FOB,  com- 
pensating the  force  (£;  then  place  the  fore-and-aft  corrector- 
magnets  so  as  to  reduce  the  deviation  FOB  to  zero,  eliminating 
the  force  S3,  and  completing  the  compensation  of  the  compass. 


APPENDIX    C. 

GENERAL  USE  OF  AZIMUTH  TABLES. 

By  the  azimuth  tables  issued  to  the  navy  the  azimuth  (Z)   is 

found  when  the  hour  angle  (t),  the  declination  (d),  and  the 
latitude  (L)  are  given;  in  other  words,  one 
angle  of  a  spherical  triangle  may  be  found 
when  two  sides  and  the  included  angle  are 
given.  Therefore,  these  tables  may  be  used 
to  find  the  position  angle  (M )  which  may  be 
desired  for  use  in  Littlehales'  method  of 
equal  altitudes  (Art.  270),  the  hour  angle 
(t)  of  an  unidentified  heavenly  body  whose 
true  altitude  and  true  azimuth  are  known 

(Art.  328),  and  the  great  circle  course  from  one  given  place  to 

another  given  place  (Art.  135). 
Let  Fig.   156   represent   the  astronomical  triangle  lettered  as 

shown;  then,  having  given  f,  L,  and  d,  to  find  Z,  we  have  from 

Napier's  analogies 

Tan*  (Z  —  ^f)=cot$fsini  (L  —  d)  sec*  (L-fd)      ^ 
Tan}  (Z  +  M)  =  cot  if  cos  £  (L  —  d)  cosec  i  (L  +  d)  /    (a) 

these  being  the  formulae  by  which  the  azimuth  tables,  now  issued 
to  the  navy  by  the  navy  department,  were  computed. 

(A)  Having  given  L,  d,  and  t,  to  find  #,  we  have  the  following 
rule:  Enter  the  azimuth  tables,  in  the  given  latitude;  at  the 
intersection  of  the  horizontal  line  through  the  given  hour  angle 
and  the  vertical  column  under  the  given  declination  will  be  found 


APPENDICES  757 

the  required  true  azimuth,  estimated  from  the  elevated  pole  and 
reckoned  from  0°  to  180°  towards  the  east  or  west  as  the  body  is 
east  or  west  of  the  meridian  (see  Art.  221). 

H.  .O.  Publication  No.  71,  in  which  the  latitude  runs  from  0°  to 
61°  and  declination  from  0°  to  23°,  each  at  an  interval  of  1°,  has 
a  different  table  for  north  and  south  latitudes.  H.  0.  Pub.  No. 
120,  in  which  the  latitude  runs  from  0°  to  70°  and  declination 
from  24°  to  70°,  each  at  an  interval  of  1°,  has  but  one  table  for 
both  cases;  but,  when  the  latitude  and  declination  are  of  a  differ- 
ent name,  the  tables  are  to  be  entered  with  the  supplement  of  the 
hour  angle  and  the  supplement  of  the  tabulated  azimuth  is  to  be 
taken  for  the  required  true  azimuth. 

(B)  Given  t,  L,  and  d,  to  find  the  position  angle  M — In  this  case 
we  will  have  from  Napier's  analogies 

TanHJf  —  #)—  cot  4*  sin  J  (d.—  L)  seci  (d  +  L)  j  m 
Tani  (M  +  Z)  =  cot|£cos£  (d  —  L)  cosec  $  (d  +  L)  f 
Comparing  formulae  (b)  with  formulae  (a)  there  is  noted  an 
interchange  of  M  and  Z,  and  of  L  and  d,  and  it  is  evident  that 
the  tables  may  be  used  to  find  M,  having  given  t,  L,  and  d;  but 
it  must  be  remembered  that  by  following  a  rule  similar  to  that 
for  taking  out  the  azimuth  in  the  cases  when  L  and  d  are  of  a 
different-  name,  we  shall  obtain  the  supplement  of  the  required  M 
instead  of  M  itself.  It  must  also  be  remembered,  in  the  case  of 
an  observed  heavenly  body,  that  the  position  angle  M  will  be 
greater  than  90°  only  when  the  declination  is  greater  than  the 
latitude  and  of  the  same  name  and  when  the  body  is  observed 
between  the  point  of  maximum  azimuth  and  the  upper  meridian, 
and  that  it  will  not  be  greater  than  90°  when  L  and  d  are  of  a 
different  name.  Therefore,  the  following  rules  should  be  fol- 
lowed: (1)  When  L  and  d  are  of  the  same  name,  enter  the 
azimuth  tables  with  *  in  the  hour  angle  column,  using  the  given 
declination  as  latitude  and  the  given  latitude  as  declination,  and 
take  out  the  value  of  M  from  the  tabulated  azimuths.  (2)  When 
using  H.  O.  Pub.  No.  71,  L  and  d  being  of  a  different  name,  follow 
the  above  rule  and  M  will  be  the  supplement  of  the  angle  taken 
from  the  tabulated  azimuths.  (3)  When  using  H.  O.  Pub.  No. 
120,  L  and  d  being  of  a  different  name,  enter  the  tables  with 
12h  —  t  in  the  hour  angle  column,  using  the  decimation  as  lati- 
tude and  latitude  as  declination,  and  the  angle  taken  from  the 
tabulated  azimuths  will  itself  be  the  position  angle  M. 


758  APPENDICES 

The  t  referred  to  here  is  the  body's  hour  angle  from  the  upper 
meridian;  when  finding  M  for  use  in  the  solution  of  equal  alti- 
tudes, it  will  be  sufficiently  accurate  to  consider  the  hour  angle 
as  half  the  elapsed  time  when  the  first  observation  is  east  of  the 
meridian,  and.  as  the  supplement  of  half  the  elapsed  time  when 
the  first  observation  is  west  of  the  meridian  (see  solution  of 
Ex.  207,  Art.  270). 

(C)  Given  Z,  L,  and  ft,  to  find  t. — In  this  case  we  will  have  from 
Napier's  analogies 

Tan  A  (t  —  M)  =  cot£Zsini  (L  —  ft)  sec£  (L  +  ft)     ^  ,. 
TanJ  (t  +  M)  =cot*Zcos£  (L  — ft)  cosec  \  (L  +  ft)| 

Comparing  formulae  (c)  with  formulae  (a)  there  is  noted  an 
interchange  of  Z  and  t  and  a  substitution  of  ft  for  d;  so  to  find  a 
heavenly  body's  hour  angle-  (t),  having  given  the  latitude  (L)  of 
the  ship,  the  unknown  body's  true  altitude  (ft)  and  its  true 
azimuth  (Z)  estimated  from  the  elevated  pole,  we  have  the  fol- 
lowing rule:  Convert  the  azimuth  into  time  and  consider  it  as 
an  hour  angle;  enter  the  azimuth  tables  in  the  given  latitude, 
with  the  azimuth  used  as  an  hour  angle  and  the  altitude  used  as 
declination,  and  take  from  the  tabulated  azimuths  the  body's 
hour  angle  expressed  in  arc  (see  Art.  328  and  Ex.  228, \ 

(D)  To  find  the  great  circle  course  between  two  places. — Enter 
the  tables  in  the  given  latitude  of  the  place  of  departure,  with 
the  difference  of  longitude  between  the  two  places  expressed  as 
time  in  the  hour  angle  column,  and  the  latitude  of  destination  in 
the  declination  column,  and  take  from  the  tabulated  azimuths 
the  great  circle  course  named  from  the  elevated  pole,  towards 
east  or  west  as  the  place  of  destination  is  to  eastward  or  westward 
of  place  of  departure  (see  page  276). 

When  the  difference  of  longitude  between  the  two  places  is 
greater  than  6  hours  and  the  value  is  not  found  tabulated,  as  it 
may  not  be  in  H.  O.  Pub.  No.  71,  enter  the  tables  in  the  given 
latitude  of  departure,  with  the  supplement  of  the  difference  of 
longitude  expressed  as  time  in  the  hour  angle  column,  and  the 
latitude  of  destination  with  name  changed  in  the  declination 
column,  then  the  supplement  of  the  tabulated  azimuth  will  be 
the  great  circle  course  to  be  marked  as  before  directed. 


APPENDICES 


759 


APPENDIX  D. 

SOLUTION  OF  THE  ASTRONOMICAL  TRIANGLE  BY 
NOMOGRAPHY.* 

The  nomogram  constructed  by  Dr.  Pesci  of  the  Royal  Italian 
Naval  School  is  a  diagram  for  the  graphic  solution  of  equations 


of  the  form  tan  a  tan  &  =  sin  c  by  the  ingenious  method  of 
aligned  numbered  points.  It  has  been  adapted  by  Lt.  Radler  de 
Aquino,  Brazilian  Navy,  to  the  solution  of  the  astronomical 
triangle  through  similar  equations  given  below  in  group  (a), 
which  are  of  the  form  cot  a  cot  &  =  cos  c. 

*  See  "  Proceedings  of  the  U.  S.  Naval  Institute,"  No.  126,  page  633. 


760  APPENDICKS 

This  nomogram  consists  of  two  parallel  scales  of  equal  length 
separated  by  a  distance  equal  to  that  length  and  so  joined  by  a 
diagonal  scale  as  to  form  a  figure  similar  to  the  letter  N  (Fig. 
157).  As  adapted  by  Lt.  Radler  de  Aquino,  CD  is  a  scale  of 
consines,  EF  of  cotangents  (limited  to  angles  greater  than  45°), 
and  ED  is  so  graduated  that  when  the  numbered  points 
corresponding  to  any  two  given  quantities  of  the  equation 
cot  a  cot  &  =  cos  c,  each  taken  on  its  proper  scale,  are  aligned, 
the  numbered  point  where  the  line  intersects  the  remaining  scale 
will  correspond  to  the  required  quantity. 

The  points  corresponding  to  cosines,  the  right-hand  functions 
of  group  (a),  must  always  be  taken  on  the  right-hand  scale  CD; 

the  points  corresponding  to  co- 
tangents, the  left-hand  functions 
of  group  (a),  may  be  taken  from 
the  other  two  scales  (the  left 
and  diagonal)  indiscriminately, 
provided  that  the  left  scale  is 
used  only  for  cotangents  of 
angles  greater  than  45°.  This 
restriction  on  the  use  of  the  left- 
hand  scale  arises  from  the  na- 
ture of  the  equation  cot  a  cot  & 
—  cos  c,  which  shows  that  both 
FIG  158  a  an(^  ^  cannot  be  less  than  45° 

at    once;    and,    therefore,    when 

either  a  or  &  is  less  than  45°  it  must  be  read  on  the  diagonal 
scale,  the  remaining  one  being  read  on  the  left  scale;  and  when 
both  a  and  6  are  greater  than  45°,  either  may  be  read  on  either 
the  left  or  diagonal  scale.  Any  two  points  of  the  nomogram,  not 
on  the  right-hand  scale,  aligned  with  0°  of  the  right-hand  scale, 
are  marked  by  complimentary  numbers  and  for  this  reason  the 
graduation  is  easily  checked. 

Let  Fig.  158  be  the  astronomical  triangle;  P,  the  elevated  pole; 
Z,  the  zenth  of  the  observer;  and  M,  the  observed  body  whose 
altitude  is  h  and  declination  is  d;  then  PZ  is  the  co-latitude.  If 
the  triangle  is  divided  into  two  right  triangles  by  a  perpendicu- 
lar Mm  and  the  parts  be  lettered  as  shown,  a  complete  solution 
may  be  effected,  through  Napier's  rules,  by  the  following 
equations: 


APPENDICES 


'61 


(a). 


cot  b  cot  (90°  —  d)  =  cos  t 

cot  B  cot  (90°  —  ft)  =  cos  Z 

cot  £  cot  (  90°  —  a)  =  cos  1) 

cot  Z  cot  (  90°  —  a)  =  cos  B 
The  following  precepts  .are  given  by  Lt.  Radler  de  Aquino  for 
the  determination,  without  resort  to  signs,  of  the  values  of  B 
and  the  quadrant  of  Z  in  the  various  cases  dependent  on  the  rela- 
tive values  of  L  and  d: 


f  /&>L:B=(90°+L)— 

d  and  L  of  same  name  J   <yu   \&<L:  J3=(90°+&)  —  L;  Z>90" 


—  90°;Z<90° 
d  and  L  of  different  name  B—  90°—  (L+6)  ;  Z>90° 

In  other  words,  in  the  first  two  cases  when  d  and  L  are  of  the 
same  name  and  t  <  6h,  90°  must  be  added  to  the  smaller  of  the 
two  quantities  &  and  L  and  from  the  sum  the  greater  should  be 
subtracted.  In  the  third  and  fourth  cases  &  and  L  are  always 
added  together;  if  the  sum  is  greater  than  90°,  subtract  90°  from 
it;  if  less  than  90°,  it  is  subtracted  from  90°. 

The  precepts  given  above  for 
determining  the  value  of  B  and 
the  quadrant  of  Z  might  be  re- 
placed by  the  much  simpler  and 
more  covenient  precepts  of  Art. 
249,  provided  the  equations  in 
group  (a)  should  be  expressed 
in  the  nomenclature  of  that 
article,  a  nomenclature  with 
which  the  American  naval  ser- 
vice is  familiar.  The  student  is 
referred  to  Art.  249  and  to  Figs. 
115  and  116  therein  representing 
the  various  positions  of  the 
heavenly  body  dependent  on  the  values  of  L  and  d  for  a  full 
explanation  of  the  reasons  given  for  the  precepts  suggested  below. 
Let  PMZ,  Fig.  159,  be  the  astronomical  triangle  projected  on  the 
plane  of  the  horizon;  then 

m  is  the  foot  of  the  perpendicular  fc  dropped   from  the 
position  angle  M  on  PZ. 

0   =Pm  is  the  polar  distance  of  m;  and  if  <t>"  =  QQ°-~  <f», 

<f)"  =  Qm  is  the  declination  of  m;  and 

A'  =mZ  is  the  zenith  distance  of  m. 


S 

FIG.  159. 


762  APPENDICES 

Applying  Napier'r.  rules  we  have 

cot  0"  cot  (90°  —  d)  =  cos  t 
coU    cot  (90°  —  fc)  :=cos0" 
cotZ  cot  (90°  — fc)  =cos  (90°  — 0')   l.(b). 
cot  (90°  —  ft)  cot  (90°  —  0')  = 
L  =  0"  +  0'. 

From  the  above  equations  and  Art.  249,  we  have  the  following 
precepts: 

(1)  0",  the  declination  of  ra,  is  taken  out  in  the  same  quadrant 
as  t  and  is  marked  N.  or  S.  like  the  declination. 

(2)  0',  the  zenith  distance  of  m,  is  marked  N.  if  the  body  bears 
southerly,  or  S.  if  the  body  bears  northerly. 

(3)  Z  is  taken  from  the  nomogram  in  the  first  quadrant  and  is 
marked  N.  if  0'  is  marked  S.,  or  S.  if  0'  is  marked  N.;   also  E. 
or  W.  as  the  body  is  east  or  west  of  the  meridian.     This  is  a 
corollary  of  (2). 

(4)  0"  and  0'  are  combined  algebraically  to  give  the  latitude 
L,  and,  when  any  two  of  the  three  are  known,  the  third  may  be 
found. 

There  are  various  purposes  for  which  the  nomogram  may  be 
used  and  the  manner  of  its  use  will  be  apparent;  but  its  principal 
use  to  the  navigator  will  be  when  he  is  seeking  (1)  h  and  Z, 
having  given  L,  d,  and  t;  (2)  t  and  d,  having  given  h  and  Z  of 
a  body  observed  at  a  given  place.  It  may  also  be  used  in  finding 
the  great  circle  course  and  distance  between  two  places,  but  these 
may  more  easily  and  conveniently  be  found  from  a  great  circle 
chart. 

The  following  is  the  key  when  seeking  h  and  Z: 

0"  is  determined  by  90°  —  d  and  t. 

0'    is  determined  by  formula  L  =  0"  +  0". 

90°  —  k  is  determined  by  t  and  0". 

Z  is  determined  by  90°  —  k  and  90°  —  0'. 

90°  —  h  (or  7i)  is  determined  by  90°  — 0'  and  Z. 
Ex.    229.— April    3,    1905,    a.    m.,    in    latitude    25°    40'    S.    and 
longitude  104°  05'  30"  E.,  the  L.  A.  T.  was  7h  32m  30s  a.  m.   (or 
t  =  —  66°  52'  30")  and  the  declination  was  4°  58'  42"  N.,  required 
the  true  h  and  Z. 

Solution  (Fig.  157):  With  90°  —  d  and  t,  find  0"  =  12°  30'  N. 
(marked  N.  as  d  is  N.),  line  IA;  and  as  L  =  0"  +  0',  0'  =  38°  10' 
S.  and  90°  —  0'  =  51°  50'. 


APPENDICES  763 

With  t  and  0",  find  90°  —  k  =  23°  30'  (line  IIA). 

With  90°  —  k  and  90°  —  0',  find  Z  =  74°.9  (line  IIIA) ;  as  0'  is 
S.  and  t  is  (— ),  Z  =  N.  74°.9  E. 

With  Z  and  90°  — 0',  find  90°  —  h  =  U°  38'  (line  IVA)  and 
7i  =  180  22'. 

The  following  is  the  key  when  seeking  t  and  d: 
90°  —  0'  is  determined  by  90°  —  h  and  Z. 
<f>"  is  determined  from  the  formula  L  =  <t>"  +  0'. 
90°  —  fc  is  determined  by  Z  and  90°  —  0'. 
£  is  determined  by  90°  —  k  and  0". 
90°  —  d  (or  d)  is  determined  by  0"  and  t 

Ex.  230.— April  5,  1905,  p.  m.,  in  latitude  20°  38'  S.  and 
longitude  90°  10'  E.,  weather  cloudy,  a  bright  star  was  observed, 
through  a  break  in  the  clouds,  bearing  N.  61°  E.  (true) ;  -)f' s  h 
25°  10';  W.  T.  of  obs.  7h  20m  40s;  C— W  5h  51m  30s,  chronometer 
fast  of  G.  M.  T.  lm  40s;  required  the  name  of  the  star. 

Solution  (Fig.  157):  With  90°  —  h  and  Z,  find  90°  —  0'  =  44° 
(line  IB)  and  0'  =  46°  S.  (marked  S.  as  the  body  bears  N.). 
From  L  =  0"  +  0'  we  have  0"  =  25°  22'  N.  With  Z  and  90°  —  0', 
find  90°  — fc  =  37°  40'  (line  IIB).  With  90°  —  k  and  0",  find 
t  =  55°  10'  =  +3h  40m  40s,  body  being  east  of  the  meridian  (line 
IIIB).  With  0"  and  t,  find  90°  —  d  =  74°  50'  (line  IVB)  and 
d  =  15°  10'  N.,  d  being  of  the  same  name  as  0".  From  the  other 
data  we  have  L.  S.  T.  =  8h  03m  50s  and  therefore  the  -X-'s  R.  A.  is 
llh  44'"  30s;  this  with  the  declination  of  15°  10'  N.  identifies  the 
star  as  3  leonis. 


764 


APPENDICES 


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8 


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